I'll be able to use this sentence a lot :)
can we use the words bathroom and washroom interchangeably? Mosdo es furdoszoba?
Usually yes. But "fürdőszoba" is usually the one you have at a home or apartment. And you can have a bath or take a shower there. A "mosdó" can be anywhere, public, private, anything. There, you can at least wash your hands.
Does this use of bejön make this a habitual task by default? Is this a way of making it imperfective, or is that reading into it too much?
I'm pretty sure just jön would be habitual. Bejön means it something with an objective/end, which would be the case if it was only going to happen once.