Scrambling jets is one thing but they still coordinate with ATC. How would they know if they are going to collide with a civil aircraft other than onboard radar?
As far as other flights, such as training routes, yes, they do coordinate with the FAA and the FAA has the authority over them. I don’t know of any law that says the FAA controls US airspace but the military has supremacy.
Your comments about the FAA and the military are correct. For example, if the military wants restricted airspace or a military operations area, it must go through the FAA and treat it like a Notice of Proposed Rule Making.