No, but you should capitalize American. If the specific branch of the military is named, it should be capitalized.
Yes. It is a noun referring to an organization. No, because military is a general descriptor. Take your pick.
It reached extremes that can not be tolerated by members of the US Military.
Yes in fact many other countries buy u.s weapons and equipment.
Tuskeegee Airmen, Buffalo Soldiers,
Yes.