2009 fleetwood providence have hot water with city water hook up but none with the 12 volt pump but do have cold with each and plenty of pressure with both
Your HWT is on the right side wall just at the very back. It is the compartment door with a 4 inch by 5 inch screen in it.
The check valve will be inside the unit though, on the upper pipe. The check valve is only about an inch long and is inline and unless you know what it is you will likely miss it.