A point that needs to be made is that the FAA and DoD ATC systems are
not completely independent and parallel systems; they're thoroughly
intertwined. For instance, Ellsworth Approach/Departure handles
civilian traffic into and out of Rapid City Regional; Wichita
Approach/Departure handles McConnell AFB traffic; military controllers
at Sheppard AFB handle traffic for the civilian side of the field
(Wichita Falls Muni) in the air and on the ground. Military sorties
are under FAA control except under two circumstances:

1. When being handled by a military ATC facility. The handoffs
to/from FAA controllers are done in accordance with procedures
established in part at the national level and in part by local FAA/DoD
working groups. Handling of civilian traffic by military controllers
is worked out the same way.

2. When operating in a Military Operations Area (MOA) or a
low-level training route. The sortie is directed to the MOA or the
entry point of the route by the FAA, then it's on its own (unless
there's a military ATC facility controlling the MOA, e.g. Nellis Range
Control) under a policy known as MARSA (Military Assumes Responsibility
for Separation of Aircraft). The FAA tries to keep civilian traffic
out of the way, but is not responsible for mishaps unless the FAA
controllers were negligent. When the sortie has flown the route or is
ready to leave the MOA it contacts Center, declares itself "no longer
MARSA" and is again under FAA control.

Emergencies are handled the same way as in civilian aviation, although
a military crew may have to justify their irregular actions to two sets
of authorities afterward. DoD flight safety regulations are heavily
based on FAA requirements and even preferences; the military comes down
hard on unnecessary violations of FARs. However, I don't know of a
case in which a military pilot who also held an FAA license was
punished by both the DoD and FAA for an action in a military aircraft.