Can the Supreme Court perpetually deny petitions for certiorari? What, if anything, prevents the Supreme Court from perpetually denying all petitions for certiorari?
No law prevents this -- the court may use its judgement on which petitions to accept. Under the current rules, any petition which four Justices choose to accept will be accepted and given a hearing. The traditions of the court indicate various reason for acceptance, but these are not legally binding. No Court to date has failed to accept a significant number of petitions for "cert".
The courts do not supersede your constitutional rights, although you may believe that you have a constitutional right that isn't actually there. This article discusses the position that "due process forbids convicting an individual of a crime unless the government proves the elements of the charged offense beyond a reasonable doubt". This standard is actually not stated anywhere in the US Constitution, but it has been assumed as an implicit meaning of "due process". It sounds like you were charged with a crime, and there is most likely an applicable statute in your state that is analogous to RCW 26.50.110 in Washington. So you have the right to a trial and the prosecution would have the obligation to prove all of the elements of the crime beyond a reasonable doubt. It also appears that you did violate the applicable law and you were willing to plead guilty, as urged by your attorney. You are correct that you don't technically have to prove your innocence, but there is a practical problem that if the prosecution provides some weak evidence that you violated the law, then the jury might decide that your failure to refute the evidence means that there is no reasonable doubt. The problem is that there is a tendency for jurors to think that the defendant has to create a doubt. States differ somewhat in how they explain the burden of proof to jurors, and you might fare better in a state where the instruction is that "you must be firmly convinced". Since the attorney seems to have said that "the constitution doesn't apply to this", this is a puzzle. I would not assume (though it is possible) that the attorney was incompetent. It is possible that he was speaking of a non-criminal matter, and it is possible that you were talking at cross purposes. There is no legal situation where "the constitution doesn't apply to this", but perhaps "that constitutional limitation doesn't apply to this specific situation". Regardless of what the attorney said, your attorney doesn't violate your rights, even if he gives you bad advice. The actual court might, and then you would have a cause for an appeal. Similarly, if the district attorney reasonably believes that you are a danger to society and is prosecuting you, that is not a violation of your constitutional rights. An improper conviction would be a violation of those rights, although it might take an appeal to get the court to recognize that fact.
The jury isn't told what the law allows. They are told to come up with a number and if it exceeds what the law allows, the judge modifies it in response to post-judgment motions.
Constitutional matters can be, and often are, decided by a single judge in a garden-variety trial court. It's just that the judge's decisions may be reviewed and possibly overturned by a higher court, one of which is the supreme court. Furthermore, district courts are bound by precedent. If a case turns on a new statute, however, the trial judge can indeed find that new statute unconstitutional without a higher court first having done so. If a panel of judges is evenly divided on whether to overturn a lower court's ruling, the lower court's ruling stands, but no precedent is set. The supreme court often has an even number of justices hearing a case, whether because of a vacancy or because a justice has recused him or herself.
There is a saying that you can't sue city hall: that is applicable here. Governments enjoy sovereign immunity, and cannot be sued for their errors of judgment unless they specifically allow it. Safety regulations are an example of a situation where the government hold all of the power and shoulders none of the responsibility. If a government forbids sale or use of a substance on some grounds (could be safety, could be economic impact, could be something about preventing the moral decay of society), and the grounds later turns to be false, you cannot sue the government because of lost business opportunities. Generally speaking, that which is not prohibited is allowed, so there would have to be an affirmative duty for the government to prevent all forms of harm. If there were such a legal duty, there is a miniscule legal foundation for suing the government for shirking its duty. That is not a completely hypothetical possibility, in that the state of Washington imposes a constitutional duty on the legislature to provide public education, and the Supreme Court has done things to enforce this duty (the contempt of court fines are up in the realm of $80 million). Since there is no jurisdiction that imposes a duty on the government to absolutely prevent all harm, you won't be able to sue the government if they fail to outlaw a thing that is eventually proven to harm someone.
Generally speaking, courts take whatever time they need to write their decision and then release it close to immediately. In cases where a judge believes she has the information she needs, she may rule "from the bench," announcing a decision and entering an order for the parties to comply, and then follow up with a written order later. The research process is fairly open-ended, but legal research databases are pretty advanced, and a good researcher can generally get his hands on the vast majority of what he'll need in very short order. But practically speaking, there are few limits on how long that process takes. A litigant who was growing impatient could seek a writ of procedendo to force a court to move faster, but my experience indicates that most attorneys wouldn't attempt that move unless a decision had been pending for at least a year with no action, which would be unusual.
There has never been a right to appeal directly, or discretionarily form a state trial court to SCOTUS. But, one can apply for a common law writ in SCOTUS with respect to a state trial court actions under the All Writs Act which is functionally an appeal, even though it is technically an original SCOTUS proceeding. But, original SCOTUS writs are extremely rare and I don't know if it has ever been used in the manner you describe. If so, it probably happened before the intermediate federal courts of appeal were created in the late 1800s. It isn't easy to do a search of cases to determine the answer. UPDATE WITH CORRECTION: After further consideration, I think that I have a better idea of what the question was getting at and want to elaborate on that. Most state courts (Utah, Texas and Oklahoma are exceptions to the general rule[1]), have either an intermediate court of appeal and then a state supreme court (sometimes given a different name such as "Court of Appeals" in New York State, in which the Supreme Court is the name of the trial court of general jurisdiction and the intermediate appeals court), or only a state supreme court from which appeals lie directly to the state supreme court. [1] In Utah there is an appeal of right in every civil case to the state supreme court and the intermediate court of appeals considers cases that it "kicks down" to the court of appeals rather than dealing with itself. In Texas and Oklahoma, there is a Court of Criminal Appeals in addition to the State Supreme Court, which operates much like a Supreme Court in criminal cases with the state supreme court that is nominally superior only rarely intervening in criminal cases (I don't know when precisely it can intervene in those states). In every state with an intermediate appeals court, there is an appeal of right to an intermediate appeals court which must be considered on the merits, and then a discretionary appeal to the state supreme court. Sometimes in these states there are a handful of exceptions. For example, in Colorado, there is a direct appeal of right to the state supreme court in death penalty and water law cases, and in municipal court and limited jurisdiction court cases (i.e. cases in courts limited to $25,000 in controversy or less), there is an appeal of right to what would usually be the general jurisdiction trial court and then a discretionary appeal from there to the state supreme court). But, there is still almost always one court to which there is a direct appeal of right. There are a small but non-negligible number of cases where what is usually a general jurisdiction trial court serves as an intermediate court of appeals of right for a lower trial court (e.g. a small claims court or municipal court), with appeal from there only in the discretion of the state supreme court. Those cases can and very rarely (but sometimes) do go on to be considered by the U.S. Supreme Court when a state supreme court denies discretionary review (in addition to being rare, few such cases are significant enough to justify the effort on the part of the parties to do so and are considered significant by the U.S. Supreme Court too), just as they would when an intermediate court of appeals rules and a state supreme court denies certiorari at the state level. But, in that case, the direct appeal still isn't a direct appeal of a "court of first instance" (i.e. a court where the original trial on the merits is held). But, there is no constitutional right to a direct appeal of right (there is only a constitutional right to a collateral attack via habeas corpus in criminal cases resulting in incarceration). The right to a direct appeal in state courts is a product of state law and state constitutions (in the federal courts it is a consequence of federal statutes). In some states without an intermediate court of appeals, there is also an appeal of right to the state supreme court. But, in other states without an intermediate court of appeals, appeals to the state supreme court are discretionary (much like appeals to the state supreme court from an intermediate appellate court and appeals to the U.S. Supreme Court). Thus, if you are in a state in which there is only a discretionary appeal to the state supreme court, and the state supreme court declines to consider your appeal on the merits, you can appeal to the U.S. Supreme Court once the state supreme court has declined your state level petition for certiorari. The actual pre-requisite at the state level in a direct appeal is that your state direct appellate options be exhausted. So, if you are in a state without intermediate courts of appeal in which the state supreme court reviews trial court decisions on only a discretionary basis, then you can make a direct appeal to the U.S. Supreme Court if you have sought review on direct appeal in the state supreme court and it declined to grant certiorari to review your case on the merits. In practice, this is still very rare. States without intermediate appellate courts are universally small in population and hence have small case loads. Most states with only a single supreme court still have at least one direct appeal of right and have had that right for most of their history. And, state supreme courts usually do exercise their certiorari power in cases that are worthy of discretionary consideration in the U.S. Supreme Court. But, in this narrow situation, the U.S. Supreme Court may consider a trial court decision on a direct appeal, without it ever having been considered on the merits at the state level. This happens rarely (although the U.S. Supreme Court not infrequently, i.e. perhaps a couple of times a year, considers a case upon which there has been an intermediate direct appeal and a state supreme court has refused discretionary review), but it has almost certainly happened sometime. It is hard to devise a legal research search that would locate such a case, however, and strictly speaking, the denial of certiorari can be seen as a state supreme court ruling that is required in almost every case.
So, as I understand the decision, it's a little more subtle than that. By default, states have sovereign immunity and can't be sued without their consent. Congress can remove ("abrogate") this immunity by law in some circumstances. They tried to do so for copyright infringement cases with the Copyright Remedy Clarification Act of 1990. However, in the present case of Allen v. Cooper, the Supreme Court held that this part of the CRCA is unconstitutional. The idea is that under the Fourteenth Amendment, Congress can abrogate state immunity when it's necessary to ensure people's right to due process, but only in a "congruent and proportional" way. Now if a state unintentionally or negligently infringes someone's copyright, that does not violate the person's right to due process, but an intentional infringement might. At the time the CRCA was passed, when Congress went looking for instances where states infringed on copyrights, they found several cases of unintentional or negligent infringement, and just a couple where they may have infringed intentionally. SCOTUS argued that to respond to this by completely abrogating state immunity in all copyright cases was disproportionate, and therefore unconstitutional. But the Court suggests in the opinion that Congress could pass a different law to abrogate immunity in copyright cases, if it were narrower. For instance, a law that only stripped immunity in cases of intentional infringement would likely be constitutional, especially if there were evidence that intentional infringement was happening enough to be a significant problem. So I think the answer is that as of right now, a state could deliberately infringe someone's copyright (e.g. by pirating software) and be immune from suit. However, Congress has the power to "fix" this, and most likely will, especially if there seems to be egregious abuse. (By the way, the decision contains an impressive quantity of pirate jokes. I guess since it's not only about copyright infringement (aka "piracy"), but actually alleges infringement of a video about a sunken pirate ship, the justices just couldn't resist.) Your "eminent domain" idea is separate from this. Seizing copies of the software wouldn't give the state the right to use them, as the software itself would still be copyrighted. The state would have to seize the copyright, and I don't know whether that is possible - it's not necessarily property in that sense. But if they did so, then they wouldn't be infringing the copyright at all (since the state itself would now own the copyright) and this case would be irrelevant. On the other hand, when a state uses eminent domain to seize property, they must as you say pay fair market value for it, and that means the market value before they seized it. So the value of the copyright in such a case wouldn't be "nothing" - it would be more like the amount a competitor would have had to pay the software maker to buy all the rights to that product. Likewise, if the state seizes your lovely house and bulldozes it to build a toxic waste dump, they owe you what someone would have paid for the house, not the value of a dump that nobody wants.