I think that if you want to overlap with the Coast Guard, then one big thing to consider is that a lot of the rescue type stuff that was important (as opposed to anti drug smuggling operations) is because the sea has rather changeable conditions. People go out under-prepared on nice day, the weather shifts, and they they're in trouble. Space doesn't really seem like it's going to catch people out like that, and under more realistic models, rescue ships are almost always going to be too far out to do anything about a ship crippled by random micrometeorite impact or something. I mean, I guess Star Trek has all kinds of weird anomalies that can mess people up, but those are generally 'out there' as opposed to near settled and patroled areas that seem safe.
Those assumptions are generally the realm of stuff that is not trying to be Space Opera.
The implications of not having stealth in space might be important here. If everyone always knows how powerful both sides are, it's possible you never get an actual war. Both sides just study the other side with telescopes, run the data through a simulation, and whichever side is going to lose surrenders before a shot is fired. But it's much easier to conceal a spaceship factory than a spaceship flying through space, if you have a planet to build it on. So the conflict switches to espionage and sabotage, trying to find out how powerful the other side is before the ships are launched, and slow down their production.
Space fights are highly mathematical and deterministic. Superior intelligence gives you a better understanding of enemy capacity, and an understanding of their knowledge of your own capacity. Conventional wisdom holds that a space battle will thus never be necessary; simply building a fleet of sufficient size and power will win the war.