Sure...except that you'd have to be convicted in a federal court for that state provision to trigger in that instance. Trump was never tried, much less charged, with "insurrection" in a federal court.

Yes - but it doesn't have to be that way. For example, you could draft a state law that says that no one can serve as a county treasurer if they have failed to file their most recent federal tax return. That's a state law requirement, but it would require a state court to determine some questions of fact dealing with compliance with federal law: was a candidate required to file a federal tax return, what was the most recent year they were required to do so, and did they do it. The fact that the federal government also has the ability to bring enforcement procedures doesn't keep the state court from making rulings on those issues.

So if the state has a rule that says that no one that has committed insurrection can run for Governor (again, for example), then a state court could make a civil determination whether a candidate has committed federal insurrection. They would be able to do that regardless of whether the federal government has filed charges or not, because they would be applying state law.