Hi. You will probably be asked to spilt these two questions.
Addressing only the bathroom one. The toilet has many names in the UK but a toilet facility that doesn't have a shower or bath in it is not called a bathroom, in general. Certainly not in the way the Americans do.
1. The bathroom is the room in which there is a bath! The presence of a bath defines the bathroom. In the bathroom, there is usually also a sink (or in AE - washhand basin) and probably a toilet and perhaps a shower. If you have a room that has only a toilet in it, that room is never referred to as the bathroom; it is 'the toilet'.
[A & B are watching TV: B stands up.]
A: "Where are you going?"
B: "I'm going to the toilet."
A: "John is in there."
B: "Oh... I'll go to the one in the bathroom."
'The Toilet' refers to the room where the actual toilet pan is (the room may or may not have a sink, bath, etc.) and the toilet pan itself.
In my house, downstairs there is a small room with a toilet and a sink - this room is "the toilet". Upstairs there is a room with a toilet a sink and a bath (with a shower in it). Because it has a bath in it, this is the bathroom.
Urgent Edit!!! I've just Googled wash hand basin/washhand basin and found that the first page of examples are from UK sites! I'm amazed! Just because I thought it sounded weird to my ears, I assumed it was AE - I have been let down by a stereotype! Seriously, I'm sorry for that! Apologies all round. Even though I have disowned it, it is British English.
Gerard Depardieu was recently inconvenienced on a plane and allegedly relieved himself down the aisle. This was reported on the BBC news where it was explained that he was prevented from visiting 'the bathroom' by the stewardess - an awkward moment I thought.
With regard to washhand basins, I was always of the opinion that it was a quintessentially British term.