You have a typical pop up or folding camper configuration. Use any pop up camper manual for most issues you may have. None of the manuals are very specific to any given model.
1) There is probably no gray water tank. The sink drain is usually connected to a lawn faucet fitting on the outside. Connect a 2 or 3 foot hose to outside and put free end into bucket or 5 gallon jug. If the drain is disconnected under the sink, put the bucket under the sink.
2) The trailer may have electric brakes. If so, a battery on the tongue is mandatory to operate the brake in an emergency. Do you have a battery box and battery on the tongue? If not you need one for the break and for the 12 volt lights inside. You may have a manual water pump or a 12 volt electric one. Look at any pop up camper for how to add a battery box and wiring. It will need a fuse or breaker.
3) I see a water fill port. Get a white potable water hose to use to fill the tank. Ordinary garden hoses will contaminate the water with chemicals and biologicals. Don’t use the white hose for other gardening uses. I also see a “city water” garden faucet fitting you can use to connect directly to a garden faucet. It by passes the water tank. Always use the white hose.
4) I see a 30 amp power cord. You said you connected it and the inside lights worked. That means you have a converter which probably will also charge the battery on the tongue.
5) The battery on the tongue usually has a self resetting circuit breaker within two feet. It is a metal can about 2 inches in size with two wires connected. It could be a fuse instead. That probably leads to an internal fuse and circuit breaker panel. The circuit breakers are for 120 volt AC circuits. The fuses are for 12 volt circuits. The converter/charger that produces 12 volts from 120 volts AC is probably all part of the same box/panel.
6) You may have a propane heater with or without a 12 volt fan. If so, you need the battery to run the fan and possibly 12 volts to control the temperature. I did not see the necessary exhaust vent on the outside for a heater, but your pictures don’t show the entire outside surface.
I wish you good luck and happy trails ahead!