Can the president veto a bill that was passed by 2/3 of each house? Can the president veto a bill that was passed by 2/3+ of each house? I mean, would it be sent back with the objections and they would have to do another show vote at least? Or is a veto-proof majority literally veto-proof? (Also, is it 2/3 of each house or 2/3 of each house's members present and voting?)
Yes, the President can certainly veto such a law. Per the US Constitution (emphasis added): Every bill which shall have passed the House of Representatives and the Senate, shall, before it become a law, be presented to the President of the United States; if he approve he shall sign it, but if not he shall return it, with his objections to that House in which it shall have originated, who shall enter the objections at large on their journal, and proceed to reconsider it. This can have a very real effect: legislators are under no obligation to vote the same way on a veto override as on the original bill. The reason the President needs to supply written objections in the first place is that it lets legislators reconsider, see if they're swayed, or see if they think this is a matter where a Congressional majority needs to be respected even if they disagree (they can change their mind in either direction). They can also get a sense of public reaction. And because the threshold for this is "present and voting," it's possible that just more legislators show up. Even if legislators won't be swayed, it still matters for pocket vetoes. That's where the President neither signs the bill nor returns it within 10 days; normally this is equivalent to signing, but if Congress adjourns in the meantime, it means the bill does not become a law. Because "Congress adjourns" is a necessary part of a pocket veto, it's impossible to override the veto (you can't do it if you're not in session). And even when this doesn't apply either, it matters for politics. Example of a futile veto: Public Law 100-4. Passed 406-8 in the House, 93-6 in the Senate. Vetoed; veto was overridden 401-26 in the House and 86-14 in the Senate (note that at least 7 Senators who voted for the bill voted not to override the veto). Example of an effective veto: While technically there was a conference report agreed to by both houses, and it doesn't seem to have had a roll-call vote (my guess is it was agreed to by unanimous consent; side note: many, many laws don't have roll-calls to check on, because they're passed by voice vote or unanimous consent), H.R.10929 from the 95th Congress was passed in the House by a vote of 319-67 and in the Senate by 87-2. After President Carter vetoed it, the House voted on whether to override the veto. The motion to override was defeated 191-206: after the veto, they couldn't even get a simple majority to override the veto of the bill which had been passed by an overwhelming supermajority. I mentioned it above, but the two-thirds threshold is "present and voting." As a general rule, any time you see a fraction of something needed for a vote to succeed in a deliberative assembly, then unless it specifies some other denominator, it's talking about the fraction of members present and voting. Relevant CRS report on override procedure.
Part of the problem you'll find is that there are so few impeachments in U.S. History (Only 21 articles of Impeachment have ever been drafted, of which only 8 resulted in convictions) and SCOTUS is so selective on cases it chooses to hear, that only one case has ever been heard and that was upheld (Nixon v. United States). In that case, SCOTUS ruled that it did not have jurisdiction to rule on the legal question before it (was the new trial format a proper trial by the senate), but did not have an opinion one way or another to suggest that SCOTUS could not review other cases that come before it. One of the reasons they also haven't is in order to have a legal case in the U.S., the plaintiff must suffer actual harm. More impeachments ended without a conviction than with either acquittal (8), resignation before trial conclusion (4), and expulsion from senate (1, and will never occur again as Congressional office holders are not impeachable following this particular case). Since no harm was caused and courts do not rule on hypotheticals, a case with actual harm (conviction) must occur in order for SCOTUS to even consider hearing the case. Nixon does not bar SCOTUS from hearing more appeals resulting from Impeachment, it only bars those relating to the manner in which the senate chooses to hold the trial.
Double jeopardy in its usual sense wouldn't attach because impeachment is not a criminal proceeding, which is the only thing double jeopardy applies to (esoteric estoppel matters not withstanding). You might recall that OJ Simpson was tried and acquitted of murder in a criminal court, and then subsequently tried and found liable in a civil court for those murders. There was no double jeopardy protections of which he could avail himself. But the constitution says that the Senate shall have the sole power to try impeachments, so for the most part we can expect that whatever they say goes. So they can dismiss for any reason they desire, in principle. The impeachment of Senator Blount is one example: the House impeached him, and on the same day the Senate expelled him under their constitutional power to do so, and then dismissed the impeachment for lack of jurisdiction (arguing that Congress members cannot be impeached; the impeachment was otherwise still relevant after his expulsion because it could result in preventing him from gaining office again). The costs here are political: in your hypothetical situation with very strong evidence, if popular opinion turns too strongly in favor of conviction then refusal to do so may cost the Senators and their party in subsequent elections. Attempts to argue arcane technicalities might not save you at the ballot box. Under existing impeachment precedent (as well as Congressional rules precedents), the courts would be loathe to get involved by default. Though if the action was sufficiently egregious (not even superficially resembling what a judge might call a trial, say) maybe they would feel judicial intervention and action was warranted and justified. But that's purely speculative.
Can Congress essentially pardon a violation of law through legislation? Yes. Congress has the power to retroactivity reduce the sentence for a crime for which someone has been sentenced. This was done most recently in the Fair Sentencing Act of 2010 that reduced excessive penalties for crack cocaine relative to powder cocaine. In the same way, when the death penalty is legislatively repealed in a state, the death sentences of the handful of people on death row at the time is often commuted legislatively. While Congress cannot impose criminal penalties on someone legislatively, which is a Bill of Attainder and constitutionally prohibited, it can single out people for special treatment in a private bill, which is a constitutional exercise of legislative power at the federal level (not every state allows private bills to be enacted due to Progressive era reforms to state constitutions). For example, private bills often eliminate the collateral effects of a criminal conviction upon a person, there is even a standard procedure for doing so, which is functionally equivalent to a Presidential pardon of the crime for that purposes (the vast majority of Presidential pardons are issued after the person convicted has served the sentence for the crime). A private bill cannot impair contract or property rights, which would be a prohibited ex post facto law. But no one other than the federal government has a legally protected interest in keeping someone incarcerated or otherwise punishing them for a crime. Crimes are prosecuted in the name of the People and victims of crimes do not have legal rights in those proceedings except as created by statute. So, this would not be an ex post facto bill or a taking governed by the Fifth Amendment for takings of property interests. And, Congress may, by legislation, determine what the federal government will do in essentially all cases where it is not expressly prohibited from doing so (some argue that there is a minimum of federal authority vested exclusively in the President in the area of foreign affairs and military affairs, but that exclusivity of power does not extend to domestic criminal justice). Basically, anything that Congress could do for everyone via a public bill, it can do for someone in particular via a private bill, unless a specific constitutional prohibition applies, and there is no such prohibition when it comes to relieving someone from a sentence or the collateral effects of a criminal conviction prospectively via a private bill. This isn't exactly equivalent to a Presidential pardon or commutation, but it is very close to one in practical effect. For a fuller, but somewhat outdated treatment of the issue, you can read this 1939 article in the California Law Review which acknowledged that legislative pardons were possible under existing law. This said, government prosecutors routinely fervently oppose any retroactive criminal legislation that reduces punishment, particularly via private laws (although some countries, such as France, have constitutional requirements to retroactively reduce the sentences of anyone currently serving time for a crime whose punishment is legislatively reduced prospectively). Private bills that constitute legislative pardons are very rare. A 2011 law review article recounts how the tool of the legislative pardon (in parallel with the executive pardon which is also used much less frequency) has fallen into disuse. Part of the decline is due to the adoption of the right to appeal from a trial court criminal convictions which did not exist in the federal system until the 1890s. Until then, the only judicial relief available from a federal criminal conviction was via a writ of habeas corpus, and that was available only on very limited grounds such as a lack of jurisdiction to conduct the trial, or the non-existence of the crime of conviction. Conviction by a jury in a court with jurisdiction of a constitutionally permissible crime was an absolute defense to the extent of the sentence imposed to a habeas corpus petition at that time. (As a footnote, the term "private bill" can be confusing. In many governments with a parliamentary system, a "private bill" means one sponsored by an individual legislator rather than the prime minister and his or her cabinet. But, in U.S. terminology, a "private bill" refers to a bill with an effect limited to one person or a small number of persons who are either identified by name or by a very narrowly defined situation.)
Shall Whenever a law or legal text uses shall there is no discretion allowed. Read it as has to when it prescribes something. If it is written in the negative (shall not), you can read it as it is forbidden or X is barred. [the] Congress of the United States [...] shall consist of a Senate and House of Representative This prescribes that the congress has to have these two parts, not more, not less. Special meetings shall be called by the president or secretary in a like manner and on like notice on the written request of three (3) Directors. Whenever 3 directors sign letters demanding a meeting, then the president or secretary has to follow the process explained before. Congress shall make no law [that does X] Note that this is a negative shall! Read it as Congress is barred from doing a law that does X. Shall obviously can have exceptions or modifiers. Such often are in modifying sentences following them. Meetings of the Board of Administration shall be open to all unit owners. Notice of all meetings shall be posted on the official bulletin board ofthe Association at least forty-eight (48) hours in advance of each scheduled meeting and directed to the attention of all unit owners. In case of an emergency, a meeting of the Board of Administration may be held without notice. There has to be a notice on the board at least 48 hours before a meeting, unless there is an emergency. May Legal texts usually use may whenever there is an option to do something. It usually marks the possibility to do something and vesting the discretion to do so in somebody. Often, they come with a minimum or maximum time an event has to occur from a deadline. The department may enter an order doing one or more of the following if the department finds that a person has violated or is operating in violation of any of the provisions of this section or the orders issued under this section: The decision if something happens is vested in the department [of Agriculture and Consumer Services]. If they decide to act, then it has to come from the list that follows the section. Special meetings of the Board may be called by the president on five(5)days notice to each Director [...] It is at the discretion of the president to call a special meeting. If they do, it has to be done with 5 (or more) days warning to the Directors. Including the other paragraph of OPs document, then it also has to be posted 48 hours before the meeting on the board, unless it is emergency.
The 25th amendment §3 says Whenever the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that he is unable to discharge the powers and duties of his office, and until he transmits to them a written declaration to the contrary, such powers and duties shall be discharged by the Vice President as Acting President. Nothing in the Constitution deprives the Vice President of his powers / duties when he is also Acting President. Alternatively, the VP can become acting President under §4, without presidential cooperation, by declaration of the VP and a majority of the cabinet. Then either the President asserts that there is no inability, or Congress decides there is. But: unless the president dies or resigns, or is impeached, the VP is Acting President and actual Vice President, and therefore breaks ties in the Senate. Note that under §4, the VP plus cabinet must declare POTUS unable, and must do it again if POTUS denies the inability. That means that there must still be a VP, who along with the cabinet re-affirms the disability.
Since electors are in fact free to vote for whoever they want (though don't usually deviate from their assignment), the branch of federal government that would be most involved is Congress. A constitutional amendment would be required, to repeal Article II Section 1 Clauses 2 and 4 and the 12th Amendment (i.e. eliminate electors entirely), and substitute a different method. Most of the work would be done by the states, in ratifying the amendment. [Addendum] It is true that it is constitutional to require a pledge of faithfulness (Ray v. Blair 343 U.S. 214). A bit over half of the states have laws requiring 'faithful voting', though the laws have not been enforced. Washington RCW 29A.56.320 may be typical, in that the law simply says "thou shall" with no mechanism for enforcement. Even with strict enforcement such as class X felony penalties, this cannot implement IRV. The number of electors is not proportional to population (there is the "plus 2 for senators" factor), and various other reasons why state-based electors cannot be morphed into an IRV-like system.
The President can nominate whomever he wants; the "advice" is formally post-nomination advice (the motion to confirm appointments is a motion "to advise and consent to" the nomination). In any event, "advice" is by definition non-binding; that's why it's not a command. However, the Senate must consent to the appointment before the officer assumes the office, so pre-nomination advice is relevant. For some nominations (like district judges), the Senators from that state can effectively sink a nomination if they're from the same party as the President and don't like the nominee; that can result in the Senators picking a short list of candidates and the President just picking someone on the list (or asking for a new list, but if he just nominates someone not on the list there's a fair chance they don't get confirmed). The Senate could decide that they will only confirm one particular person for the post. The President can nominate someone different. That's a political fight to be solved by gamesmanship and negotiation, not something that has a legal resolution.