1. it's working properly if it is outputting water out the side, 110V is showing on the power panel, and you can also tell just by the sound. If it is a constant humm, not cycling up and down. If it is cycling you may need to add more load to it.
2. Yes, there is a sea cock for both the generator and air conditioner. I'm not sure where they are for you, but there are plenty of 270 owners on here that should chime in. If I were to guess, maybe under the aft cabin cushions there is an entry into the bilge area for the a/c one.

Genny one is probably within the ER relatively close to the unit.
3. Fuel consumption depends on the load, but I would say around 1 gal/hr fully loaded.
4. Yes, you can run it many hours. I've seen them ran for entire weekends. Just make sure to run it a few minutes without a load before shutting off, so it can cool down a bit when you are done.
Good luck and enjoy using the genny! It sure makes boating on the hook very pleasant!