IMNAL, but on your private property you are just fine. Now that doesn’t make it ok to stand in your front yard waving you gun like a lunatic, but proper responsible ownership is not a problem. Also it is ok to legally transport your gun past a school while heading to another legal place (like a shooting range, or even a friends house). All your local laws still apply, but transporting within 1000ft (local laws define legal transport) and owning on a private residence it legal. As for Fed law vs. state law the more restricting law is what is enforced, and if they contradict each other Fed law wins.
Edit (even though fed law wins, you may still end up fighting it in court to prove that point)