Many training schools are used by several branches. The cop school at Lackland is used by the Air Force (duh) and the Navy. Military Police investigator school is conducted by the Army and attended by the Air Force. I currently work near a SeaBee training base and go on it frequently. There are Army and Air Force attending the training at this base. An Air Force base I work near and go on frequently also trains Navy and I believe I've seen some Army there as well.
The Marines are part of the Navy so they interact often, because they are same branch, although many Marines I know say they are in the mens section. I also include the Marines when I refer to the Navy, but that is mostly to irritate the Marines. (I love to mess with those guys)
As far as the rest of your questions, the answer is it depends. It depends on the base. When I was stationed in Turkey we were co-located with the Army. The Air Force acted as their personnel section to a point. The Air Force also performed all of their support functions. We used to joke that one Army general was the only reason 900 Air Force members and their families were at the location. At this same base the Navy provided all of our household goods shipments. At another base I was stationed the only support the Navy provided was for major medical issues, but we did patrol the Marine reserve base just outside our gate. The Air Force only had a small clinic requiring support form either civilian facilities or the local Navy base, which had a large hospital. There was no interaction with the Army.
But when a long train of abuses and usurpations, pursuing invariably the same object, evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new guards for their future security.
Translation for the intellectually challenged: If the government screws the people too much, it is the right and duty of the people to revolt and form a new government.