Dual sovereignty: Which sentence takes precedence? If one is tried for the same offense in both state and federal court, and is convicted in both cases, then which sentence is enacted first? For example, Dylann Roof got multiple life sentences at the state level, but the death penalty at the federal level. Obviously if the state sentence is enacted first, the death penalty would never be enacted, since he would first spend all his time in a state prison until he dies of natural causes.
There is no absolute rule in such cases. It is often a matter of negotiation between the state and federal authorities, and failing agreement, a matter of which authority has the prisoner in custody. Often the question of which crime is more serious or carries a longer sentence is an issue in such negotiations.
Under U.S. law, double jeopardy prevents you from being charged with the same charge twice, and also from being charged with any offense which is a lesser included offense of the charged offense, or a charge so substantially similar that for constitutional purposes it amounts to the same crime. Basically, the test is whether a prior acquittal would be inconsistent with a new criminal charge. For example, even though there is an additional element of the crime of murdering a postal officer to the crime of murdering someone on federal property, double jeopardy would probably bar a retrial of a murder on federal property case simply because the victim happened to be a postal worker and that element wasn't charged in the original indictment. This is because the acquittal of the first murder charge would almost always imply a jury determination that a murder didn't take place which would be inconsistent with a murder of a postal worker charge. On the other hand, a trial on a murder charge would probably not bar, for example, a trial on a burglary charge (which at common law involved trespassing with an intent to commit a crime), even if the burglary charge arose from the same conduct. This is because an acquittal on a murder charge isn't necessarily inconsistent with the existence of a trespass, or with the intent to commit some crime other than the murder for which the defendant was acquitted. But the exact way that the line gets draw is tricky and while what I have described is a good general summary of the cases interpreting the double jeopardy clause, it isn't a perfect one. This issue has been litigated many, many tines over the years, so there are a lot of cases that are squarely on point addressing specific fact patterns in precedents that are binding case law that are not always a perfect fit to the general principles. In these circumstances, the binding case law is going to control, at least until a court with appellate authority over the court whose case established the precedent in question decided to overrule a prior precedent from the lower court, or in the case of U.S. Supreme Court precedents, until the U.S. Supreme Court revisits one of its own prior precedents as wrong decided or wrongly interpreted, which happens now and then, although it is a rare event.
It depends on what the jury said, and if it's criminal or civil. In criminal cases, the judge may almost never set aside a verdict of acquittal. There is a single case in the US in which this happened, and it was a bench trial (no jury). That case featured the defendant bribing his trial judge; the Seventh Circuit held that he was never in jeopardy due to the bribe. As far as I can tell, that's the only one. There have been no cases that I can find of a jury's verdict of acquittal being overturnable. Judges can poll the jury to make sure they're unanimous (at least in federal court), and if they aren't then it's a mistrial, but that's because the jury was never in agreement in the first place. On the other hand, a judge has several ways to enforce an acquittal. In federal court, for instance, the defense can move for a motion of acquittal either before or after the case goes to the jury. If the motion is granted before the verdict, double jeopardy applies to retrial. If it's granted after a conviction, then the judicial acquittal can be reversed on appeal, possibly requiring a new trial. Before the verdict is returned, the judge can declare a mistrial. After the verdict is returned, it's too late for that. In civil cases, things are more complicated: double jeopardy does not exist there. There, there is a notion of a judgment as a matter of law: the judge determines that, based on evidence presented, no reasonable jury could possibly find the other way. This can happen before or after the verdict, and is appealable.
If you are convicted you can be retried (indeed, on appeal, you asked for a retrial). Double jeopardy prevents retrials in cases of acquittals and some mistrials, not convictions.
Criminal conviction by a judge after acquittal by a jury would violate the Double Jeopardy Clause of the 5th Amendment. A criminal conviction involves both a finding of fact and the application of the law, and a judge (unless this is a bench trial) doesn't find facts, he makes judgements of law. Overturning an acquittal after a jury trial would thus be in essence a whole new trial (moreover one where the defendant was not given a renewed opportunity to defend himself).
The jury never finds there was “no crime” They either find that the state has proven that this particular defendant committed this particular crime (guilty) or they have not proven it (not guilty). Another jury at another time may find the opposite - this does sometimes happen where a guilty verdict is appealed and the appeals court orders a retrial. For another defendant charged with a different crime (e.g. accessory to the first crime) before another jury, the result of another trial is both irrelevant and inadmissible.
The Supremacy Clause. U.S. Constitution, Article VI, Clause 2: This Constitution, and the Laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the constitution or laws of any state to the contrary notwithstanding. [emphasis mine] This is also known as the "Supremacy Clause" of the U.S. Constitution. It prevents any law of any state from acting contrary to the federal constitution. A comment correctly cites multiple cases that use this clause, in part, as the basis for SCOTUS' authority to review State Supreme Court decisions. So in your hypothetical, the federal issue in play is the clarity of the federal constitution which supersedes state law.
Murder Which is the unlawful taking of a life with intent to do so. However, the doctrine of self-defence can make killing lawful: A person may use such force as is reasonable in the circumstances in the prevention of crime, or in effecting or assisting in the lawful arrest of offenders or suspected offenders or of persons unlawfully at large. if the prosecutor is of the opinion that the force used "is reasonable in the circumstances" they may not lay charges. If they do lay charges the judge may decide that there is no case to answer before going to trial if self-defence applies. If there is a trial this will probably be the strategy the defence employs and they may or may not be successful. If convicted the penalty is life imprisonment. Also, there is no UK law: there is the law of England and Wales, the law of Scotland and the law of Northern Ireland.