Yes, it is slang and means NO with an attitude or as you say No way! I wouldn't recommend using it.

Yes, it can mean "not likely", "no", "no way", "you must be kidding". It also has a literal meaning, of course, of please do something for me. Like a lot of English (any language, perhaps), you need to "read" the context and the speaker to be sure of the meaning. https://www.urbandictionary.com/define.php?term=Do%20me%20a%20favor

Yes you are right. British people can be very sarcastic and this is an example of sarcastic humour. There would be quite a strong intonation in this phrase as opposed to it's typical use.