How does binding precedent work with juries in trial courts? How is a jury in a federal trial affected by binding precedent such as that of a SCOTUS decision? If the jury is selected from ordinary people, they obviously are likely to have no familiarity with legal matters. How is their ability to render a verdict influenced by binding precedent? Can a jury render a verdict that contravenes with binding precedent?
Can a jury render a verdict that contravenes with binding precedent? Yes, but (if it is a guilty verdict) it will be overturned on appeal. The appeal court will say that the jury's verdict was not one which any reasonable jury could have made on the basis of the evidence and the instructions in the case. As explained in Dave D's answer, the trial judge will have explained the law, including the binding precedent, to the jury. So if we get to this stage then the jury is disregarding instructions. The above process is the remedy. You may notice a 'gap' where the jury can unlawfully acquit a person and there's nothing anyone can do about it, because of the rule against double jeopardy (that is, you can't appeal against an acquittal). This is called jury nullification, because the jury has nullified the relevant criminal law by acquitting an offender.
The phrase "reasonable doubt" was formed hundreds of years ago, and does not hold any mathematical or probabilistic meaning. It is for each individual juror to decide for themselves what constitutes "reasonable doubt", and whether the evidence presented to them has crossed that threshold. EDIT for extra clarity: As stated above, the definition of "reasonable doubt" is intentionally vague*, and left to be decided by each juror for themselves on a case by case basis; as such there is no single rule that can be applied to jurors (also note barring accepting a bribe, a juror cannot be legally sanctioned for their conduct as a juror, nor their vote, regardless of the evidence before them). So one juror might judge by P(A|B), another might judge by the defendant's appearance, another might judge by the majority of their peers (so that they can go to a ball game that evening, such as in the film 12 Angry Men), another might disagree with the law (see: jury nullification) and so vote not guilty on that basis, and another might bow to social pressure and convict despite overwhelming evidence that the defendant is not guilty (for example, at the end of To Kill A Mockingbird). A prosecutor cares about convincing the entire jury that the defendant is guilty(outside of Oregon and Louisiana, where only 10/12 vote is needed to convict, so the prosecutor only cares convincing 10 jurors). The defense only cares about convincing a single juror (or three in LA or OR), although more can be useful to prevent a mistrial. The defense (in theory) should not care whether or not the defendant is guilty. *The origin of reasonable doubt was in Britain, where certain jurors would refuse to convict, despite any evidence, due to religious prohibitions of "Judge not, less ye be judged".
In any court, there will be situations where a judge has discretion to make some decision. The judge might have to decide "did X meet the burden of proof", and due to the situation two reasonable and competent judges could come to different conclusions. You couldn't blame either for the decision, even though they would make opposite decisions. On the other hand, a judge might make gross mistakes. The judge might decide "X met the burden of proof" when this is clearly a mistake. That's what the appellate court is interested in. An appellate court checks whether the judge made mistakes that a judge shouldn't make. So in this situation, the appellate court doesn't decide whether X met the burden of proof. The appellate judge decides "did the trial judge make a decision that a trial judge shouldn't have made". An appellate judge might think to himself or herself: "well, I would have decided differently, but this trial judge’s decision was one that a reasonable judge could have made", and if that is what he or she thinks, the original decision will stay intact.
If a law is struck-down as unconstitutional, but all the precedent used to find it unconstitutional gets reversed; what becomes of the law? In U.S. law, the law has effect again, unless it has been amended or repealed in the meantime. Is it totally dead, needing be passed anew? In the U.S., no. It is not totally dead. It is merely dormant. It stays on the books and legislators may decide not to repeal it as a political statement. It also might be considered for interpretive purposes when construing another part of the same law. For example, the meaning given to a phrase in an unconstitutional part of the law might be applied to a different part of the law that is constitutional. Can the judiciary be asked to reinstated, after which point it can be used again? In the U.S., any court can determine that a law is unconstitutional, but the extent to which that ruling is binding precedent on other courts or other parties than those to the case before it depends upon the court in question and upon the doctrine of collateral estoppel (a.k.a. issue preclusion). For example, the legal fight in the U.S. to hold bans on same sex marriage to be unconstitutional was fought in and resulted in ruling in dozens of courts at the trial court and state appellate court, and federal intermediate appellate court level before a uniform ruling was established by the U.S. Supreme Court. Further, even if the issue arises in another case where there is a controlling precedent, attorney ethics permit an attorney to make a good faith argument for a change in the law to any court, so if there is some good faith argument for doing so, the attorney can push that the issue be reconsidered. Of course, usually the answer from the court will be "no." Or can it just be enforced again without any formal process; so long as nobody sues and gets it killed again by a lower court? Sometimes government officials enforce laws that have been held unconstitutional, either because they aren't aware of the relevant court decisions, or because they think that their facts are distinguishable from those under which the law was held unconstitutional (which sometimes happens on an "as applied" basis rather than on a "facial" basis that applies to all cases), or because they think the judge before them might rule differently despite the precedent. Also, would the answer differ according to country? If so, could you please give me some examples of countries handling this differently. Yes. Many countries with legal systems based upon the legal system of countries of continental Europe like France and Germany and Spain, which are called "civil law" countries have a very different process of handling unconstitutional laws, as does the European Court of Human Rights and the highest court of the European Union. In Germany, for example, questions of the constitutionality of a law may be raised only in the Constitutional Court and not in other courts. This ruling is usually final. And, unlike U.S. courts, the Constitutional Court can rule a law unconstitutional during the legislative process, rather than in connection with an actual case or controversy relating to the law taking effect (in which case the law never gets on the books in the first place). I don't know what happens when the Constitutional Court declares a law unconstitutional. I do know, however, that in the case of the European Court of Human Rights and the highest courts of the E.U. that one of the usual remedies will be an order directed at a member state to amend its statutes to remove the offending law, with sanctions imposed if the member state fails to do so. Obviously, once such a law is repealed in this fashion, it would have to be re-enacted to take effect even if the precedent holding that the law was unconstitutional was undermined.
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.
It is binding precedent for lower federal courts in the Ninth Circuit. They are required to follow it. It is persuasive precedent for the other circuits and for state courts. They may be persuaded by the reasoning and will consider the fact that the Ninth Circuit held as it did to be one factor in their decision-making, but they can make a different decision if they want to. Some courts are also more persuasive than others, although this is usually not explicitly acknowledged in written opinions. For example, state courts in the Ninth Circuit are likely to give more weight to a Ninth Circuit opinion than they are to give weight to a decision from another circuit. Cases from the Second Circuit are more likely to be persuasive than cases from other circuits, because of its reputation. Cases from the Southern District of New York are more likely to be persuasive than cases from most other non-local District Courts. There is also a personal reputation function that comes into play when looking at persuasive decisions. Some judges have a better reputation or a better reputation with a particular other judge, and their decisions may be considered more carefully. This is not explicit, but it means a good lawyer will mention the judge's name when a case in support of his position was decided by a well-regarded judge. Finally, the Ninth Circuit's holding would be persuasive precedent for the Supreme Court. You could write law review articles about this, but to dip one toe in: they may be persuaded by the reasoning and it matters to their function of providing unifying law, so especially during the process of applying for a writ of certiorari, they will care what different circuits have held on an issue and which circuits are going which way. But they are not bound by the circuit courts, and will overturn all of the circuits if they think that's the right decision. They did that a while ago with a statute about what it meant to use a firearm while committing a crime, for example.
The appellate court gets the trial transcript which includes all the testimony (and exhibits or descriptions/photos of them) and the trial court's rulings. Also, each side presents an appeal brief, which will generally include a statement of facts, pointing out ones it thinks favorable to its views. In civil cases a jury may be asked/instructed to give yes/no answers to a series of questions, but this is not normal in criminal cases. An appellate court can (and sometimes does) rule that there was not sufficient evidence to permit any reasonable jury to convict; to that extent it can rule on questions of fact. But, as the comment by ohwilleke mentions, an appellate court does not hear new evidence. It can overturn the conviction and send the case back for a new trial, and in such a new trial additional evidence may be introduce at the trial court level.
There's nothing illegal about the media discussing your case. In fact most media outlets don't hold back many details about the accused, because it's all public information that anyone curious about it can get from the court clerk's office for free and with no reason specified. If jury selection hasn't started yet, then asking each juror about what they might know about the case is a pretty standard question that any competent defense attorney would be asking, even without knowledge of media coverage. If they already know about the case and say they can't put that knowledge aside when rendering a verdict, they are almost always dismissed. If you're concerned your defense attorney isn't aware of the media coverage, that's something you need to bring up with them. If jury selection already took place, and nobody bothered to ask them if they had prior knowledge of the case, then there's pretty much nothing you can do. A judge might ask the jury if anyone's seen the coverage, but it usually results in nothing happening. The media talking about your case is not grounds for a mistrial; it's rarely even grounds for a change of venue. Getting a mistrial would require you to prove a specific juror engaged in misconduct somehow.