"Where is the restroom?"
If it's the restroom of your host, yes. You're expected to be polite about stuff of others and modest about your own stuff. If a guest would ask you where your restroom is, you should direct them to your "humble restroom" and never preface it with an お.
If you're asking this question it's almost certainly to someone you don't know, where you would probably use this style of nouns for most things.
Yes. トイレ obviously is a cognate of our word for "toilet," but refers to the room, not just the actual toilet itself.
Yep it's common in British, Australian, and NZ English to use "Toilet" for the room.
I had some trouble remembering "てあらい" at first, but the kanji provide a helpful memory link: "手洗い" is essentially "hand" (手) and "wash" (洗い), both words already learned in the Duo tree.
Does anyone know whether it is more proper to ask the location of お手洗い rather than トイレ when, for instance, you want to wash your hands before a meal?