Do I discard Unguided Rockets like normal after use or does it hang around until the ship is destroyed? All other missiles/torpedoes say to discard in order to use but this one doesn't.

Unguided does not have any text that tell you to discard it. Its permanent, hence why its so limited (2missiles, only your focus can mod it) because its endless use.

It doesn't say to discard it so there is no need to discard it.

FINALLY, some ordnance that keeps on being useful. Now if it just had a little more bite to it...