Good to have you on the forum. My first response to the bathroom faucet is that sediment and scale buildup from the water heater has clogged up the screen at the faucet nozzle. Unscrew the end of the faucet and check to see it this is the case. Do you have the single handle faucet where you push it back to increase water flow, rotate to the left for hot water and to the right for cold water? If the screen is not clogged then the valve may be. There is a setscrew at the back of the single handle faucet. Use an allen wrench and insert it into the setscrew. Rotate counterclockwise looking at the setscrew from the top. This should loosen the handle enough to remove it. You should now have a collar you loosen and the valve can be removed. Clean it out and install everything in reverse order.
What concerns me is the water coming out from under the trailer. Do you have the rear bathroom which has a rear compartment underneath the rear window? If so, open the compartment and look to the left. You should see copper pipes, two valves and the regulator system. Could it be that someone winterized and left the valves open? If so, water would flow out of these valves, down a short pipe and into the bottom of the trailer. Look at the area after someone turns the water on to see if water is flowing from this area. I am describing what the system looked like on my '77 Excella 500 31' rear bath unit. Hope this gives you some idea of what to look for.
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