In general, if the generator is connected and running *AND* you have a good converter/charger, the water pump, lights, exhaust fans etc. should operate while the battery is being charged. Do any of these other things work, and just the water pump is down? Or is your entire 12v
Problem 2 is difficult to answer, since it seems like a PO has taken some shortcuts in adding a graywater tank. As originally built, your trailer could accumulate a small amount of graywater in the space between the black tank dump valve and the sewer cap. It sounds like you're saying the PO has attached some connection from there to the added gray tank. Since you specify that there are two valves, I wonder if the PO put in 1 valve to close off the graywater coming from your drains into the sewer pipe, and another to let the water stopped by the first valve flow into the gray tank. If that's the case you want the valve to the sewer pipe close and the valve to the gray tank open. Once you get into "custom" installations, figuring out what the PO was thinking can be a bit of a crapshoot.