Iirc I had to check in my checked bag before going through security. Also had no problem with putting my Nintendo Switch and controller in my checked bag.

Although I only flew within the EU, so different regulations may have applied than in the US.

Afaik you should be ok as long as you have it in your checked luggage, not your carry-on bag

Only mechanically. You could technically remove the small piece of plastic at the end of the slot, and still put a x16 card in a x1 or x4 slot, should work anyways. Some mainboards even have open-ended slots directly instead of closed ended. Haven’t done it and wouldn’t recommend it, but it would technically be possible.