Is there a way to find out if my living room ceiling has insulation?
Interior walls aren't typically insulated.  But to be sure you could just drill a small hole (make sure you're not on a stud) and if there is insulation it should come out on the drill bit.  But be very cautious if there is plumbing in the area, you don't want to pierce a water line or drain pipe.  Other than that, this won't hurt anything since the ceiling is already water damaged.  Just putty over the hole before you repaint.

Though, if there is significant water damage you should probably replace the section of ceiling anyway as it may be weakened and/or also have mold growing on it.