How would double jeopardy work? Does bringing a case through federal courts prevent being tried for the same acts in a state court?
As
@W.D.Clinger has just noted, generally no. Dual sovereignty is not considered a violation of double jeopardy. However, historically this was only rarely ever a problem. Congress has enumerated powers, meaning it is limited in what kinds of laws it can make. Thus historically the federal criminal code was minimal and did not encroach upon state authority. In contrast, states have police power, which political science defines as the essentially unlimited power to regulate conduct. Thus most criminal offenses were historically state charges. This meant that there was only a little overlap between what you could be prosecuted for under state law and what you could be prosecuted for under federal law.
Famously there is no federal criminal law against ordinary murder. By "ordinary" I mean the killing of one person by another without any special circumstances. There are only murder-adjacent crimes, such as murdering a federal officer (i.e., special circumstances regarding the victim) or using a firearm in the commission of a felony (i.e., federally cognizable means or modes).
With regard to the statute of limitations, I'm not sure what the argument is there. Minnesota may bring charges at any time irrespective of who is running the federal government, and it is most likely under state authority forbidding murder that charges would be brought against the officer.
If the officer wishes to escape state charges by arguing that he is immune as a federal officer, he would remove the case to federal court and have to convince a federal judge that the circumstances alleged are within the proper and lawful exercise of his office. If convinced, the judge has the authority to dismiss the state charges. In some cases, this determination requires extensive briefing and testimony, in which case the
federal judge presides over a trial involving
state law, which is a little weird. The typical defense is via the Supremacy Clause, which nevertheless requires a showing that the officer's actions were authorized by statute, and were necessary and proper to the execution of his duties. This has to be particularly balanced against the specific
actus rea requirements of the state law.
