I'm assuming that you have a water tank and not a tankless water heater.
Check to make sure that the bypass valve --the one between the cold water inlet and the hot water outlet from the heater tank -- is closed. It sounds as if it might have been left open. In the open position, the handle is parallel to the pipe, in the closed position it's perpendicular to or "blocking" the pipe.
This valve is opened, and the input and output valves closed, to prevent antifreeze from going into the tank during winterizing. If it's left open, cold water from the input side will be able to mix with hot water coming out of the tank, giving you only warm water at the hot faucet.
On the bathroom issue, you might remove the screen from the end of the faucet and make sure that it's not blocked. Close or cover the drain first to be sure no small parts go down the drain.