What happens when a president is part of a mafia that violates the law in other countries? In his confession, one of the individuals arrested in FIFAgate (corruption, bribes that were paid through United States financial institutions) mentioned former President of Argentina Cristina Fernández de Kirchner, now Senator, and other former ministers. What can the United States do now? Can the United States arrest ministers, senators or presidents from other countries?
Defendants in the US would be charged with racketeering, not bribery, since government officials were not bribed. The DoJ indictment against Webb et al. is here: most of the defendants are not citizens of the US, though none are listed as being government officials. DoJ could certainly seek an indictment of ministers, senators or presidents of foreign countries. If said official were in the US on an ordinary passport, they could be arrested. They also might be arrested by e.g. Argentinian policy and extradited to the US, but whether that would be legal depends on the country (some countries don't extradite their own citizens; there would have to be an extradition treaty between that country and the US). It is inconceivable that any nation would hand over a sitting president because of an indictment by the US, and generally unlikely for any government official, but the official could be locally deposed first. But whether a government would do this is basically a political question, not a legal one.
When you are in another country, you are subject to their laws - you may be arrested and go through the due process as defined by that countries laws. This potentially means anything from a fine, to incarceration or deportation or even execution, depending on the local countries laws. A good example is the caning of American citizen Michael Fay in 1994 by the Singapore authorities, as a judicial punishment for vandalism, or the case of Swiss citizen Oliver Fricker, who was also caned in 2010 for vandalism.
Impeachment of a president does not on conviction automatically disqualify the convicted party from becoming president again. However, after conviction, the Senate can vote to add to the punishment of removal from office "disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States." This requires only a simple majority vote. Does impeachment by the House of Representatives, with or without subsequent removal from office by the Senate, actually disqualify one from being re-elected to the same Presidential office? See above. It requires impeachment by the House, conviction by the Senate, and a separate vote by the Senate to impose the punishment of disqualification. It's conceivable that the Senate could disqualify the convicted party only from becoming president, though it looks like in the two prior instances where this punishment was imposed it was the broader disqualification. But if a person disqualified only from being president is in an office that would normally be in the line of succession, that person is simply omitted from the line of succession. This happens routinely with naturalized citizens, and there's no reason to think it would be any different for a former president who had been disqualified only from the office of the president after being convicted on articles of impeachment.
They are not given independence from statute. This clause just says that conviction is not the end goal of the prosecutor. If in light of the evidence, the prosecutor comes to believe a person is not guilty, they are not to proceed with the prosecution. They must not hide exculpatory or mitigating evidence in order to get a conviction.
One does not lose legal rights by becoming a government official, so POTUS retains the right to sue for defamation, hold copyright, sue for trespass or breach of contract, and so on. The standards for defamation change when one becomes a "public figure" (you have to show "actual malice"), but this is much broader than being a government official. Anything that is a "work of the US government" is not protected by copyright, so presidential decrees, as government works, are not protected by copyright. I do not know of any state where one legally loses publicity rights as a function of being famous, or being an elected official. California Civil Code §3344 spells out the right of publicity in that state, which says that anyone who knowingly uses another's name, voice, signature, photograph, or likeness, in any manner, on or in products, merchandise, or goods, or for purposes of advertising or selling, or soliciting purchases of, products, merchandise, goods or services, without such person's prior consent can get sued. However, there is a "fair use" escape clause: For purposes of this section, a use of a name, voice, signature, photograph, or likeness in connection with any news, public affairs, or sports broadcast or account, or any political campaign, shall not constitute a use for which consent is required under subdivision (a). The law doesn't say exactly what constitutes a "political campaign" or "public affairs broadcast or account", but since politicians get caricatured in the papers all the time, with no requirement for consent, it is highly likely that the use you point to would be found to be part of a "political campaign" or "public affairs account". Additionally, under the First Amendment, you can criticize a government official, and that right is not limited to just critical words. It is obvious that the things on sale are basically criticism of POTUS, and you can't use the law to suppress such criticism. Accordingly, one could also criticize Tom Cruise (not a government official) using his likeness on such an object. However, one cannot exploit his image to sell perfume.
Well, moral obligations are not laws, nor sometimes even moral obligations. Some laws are based on what some courts and legislatures think are moral obligations. We think cannibals have a moral obligation not to eat people; cannibals think non-cannibals are fools for passing up a good BBQ. As for a leader's moral responsibility for millions of lives, we can take as extreme examples Mao, Hilter, Stalin, Pol Pot, etc., that there are no obligations. The law that could apply in the case of a POTUS who does not "faithfully execute the Office of President of the United States..." (oath of office for POTUS) is that of "High Crimes and Misdemeanors", the grounds for most impeachments. But "faithfully executing the Office of President" has no moral obligation; it has legal implications, that's all. High crimes or misdemeanor are legal definitions, not moral, and depend on previous cases of what a crime or misdemeanor is. Insobriety can be illegal; there are laws on the books concerning public drunkenness. But in the contest of POTUS (one who is hopefully not passed out on the sidewalk in front of the WH), it remains to be seen if insobriety is a high crime or misdemeanor. That would be up to the House Judiciary Committee and US House of Representatives, which by Constitutional powers handles impeachment proceedings. The Twenty-fifth Amendment outlines who succeeds the president due to "Inability to discharge the Powers and Duties of the said Office," but it does not state who has the power to declare a President incapacitated. It's possible that a POTUS who incapacitates himself with alcohol is guilty of a "high crime or misdemeanor." But how drunk do you have to be to be incapacitated? Drunk enough to think a Game Boy is the nuclear football? Again, that's up to the House. Some past presidents have arguably been functional alcoholics (or functional recovered alcoholics.) But incapacitated? That can be subjective when it comes to the application of the law. And morals have little to do with it, unless those morals have a basis in that same law. Update 12/04/16: Some federal judges have been impeached due to drunkenness: http://www.fjc.gov/history/home.nsf/page/judges_impeachments.html and http://www.crf-usa.org/impeachment/high-crimes-and-misdemeanors.html
Criminal charges are filed and prosecuted by the government, on behalf of the public, and there is no requirement for approval or cooperation by the victim. As a policy matter, a DA may decide to not charge a person in case the victim is unwilling (though less so in cases of domestic violence), perhaps because of the widespread impression that the victim has to "press charges" (which indeed they would have to do in the case of private prosecution, which is no longer allowed in the US). The alleged victim's reddit exchanges on the topic are here. An uncooperative victim does not make a good witness, even if they are compelled to testify. The police statement that they cannot pursue an investigation should not be interpreted as a statement of law, it's probably a statement of policy and practicality.
This is an interesting hypothetical. In this scenario, Country Z does not have jurisdiction to enforce such a law on foreign nationals, unless Country Z has an extradition treaty with Country A. Generally, however, these types of laws would never be enforced as they are egregious abuses of government, and could possibly be elevated to the International Court of Justice if Country Z actually charges any individuals with such a crime. In these instances, however, war is a very unlikely scenario, since this would often be expensive and any escalation would most likely be small skirmishes that would lead to an eventual ceasefire, with the encouragement of the international community, without the involvement of UN Peacekeeping troops.