very good questions. Nav systems only help you from grounding the ship, not hitting another ship. Does your car GPS tell you anything about the big semi barreling down on you at 70 mph, no it doesn’t. Neither does the Navy’s GPS, it tells you where you are, not where the 50,000 ton tanker is. Naval vessels are not required to have AIS. I do not know if our ships have AIS or not. Surface search radar, tactical data systems computers, compass pelorus, eyeballs and binoculars are the tools to avoid a collision with a ship at sea. The other questions you asked will be very similar to the questions asked of the OOD, JOOD, CICWO by the Navy’s investigating officers.
i don't know about naval combat ships, i bet they don't broadcast AIS, but i have been at sea many times and rarely recall seeing a large tanker of container ship not using it. so a naval vessel should be able to at least receive, but i don't know.
on the radar scan, a ship the size of a container vessel looks like a giant bratwurst. you could read a book by the return signal.