Does the Coast Guard have submarines?

Not that I know of, but I guess elements of the USCG can be seconded by the USN to assist in a law-enforcement capacity? it makes sense that the maritime forces get to play together though.

Also guess it would not violate the posse commitatus (sp?) act because this is off shore and not involving US citizens.

This is correct. The USCG often rides along with the Navy in order to conduct blue-water LE operations, as the Navy and DoD are barred from LE by Posse Comitatus.

Navy has vastly more maritime resources than the USCG and will pass info, but doesn't board ships or engage in pursuit on it's own.

It's actually a good system, as the difference between Defense and LE is important to maintain.

Depends on where. International waters? No problem. Close to shore? Need LE backup to be legal. Navy VBSS teams can and do run interdictions on the regular in open seas.

The movie Sicario is basically premised off this idea of getting a LE Officer onboard for legality’s sake.

Nope

By "hunt" he means "they're on the list of things we listen for and track and when the sonar guy finds one we pass the tip to the guys on land who'll pass it to the coast guard"

A submarine isn't gonna blow its cover to conduct what's basically a wet traffic stop except maybe in some edge case circumstances and even if they wanted to there's a bunch of other ancillary reasons not to do that.

By hunt I meant a specific deployment