Generally you can call any water outlet a 'faucet'. Many people refer to it as "tap" as well, you can commonly hear the term "tap water/ water from the tap" but technically, the tap is not the faucet itself, it is what you use to regulate the water output with. Some other terms that are commonly used are: "tub spout" "shower head" :)

I will leave it to an American English speaker to answer this - we don't use the word 'faucet' in British English - I think it means a tap, but I don't know if they use it to refer to the outlet for water in showers and toilets as well.