昔、アメリカ人女性を日本人の家に連れて行った時、そこのご主人に対して Do you speak English?と言ったので、あれっ？と思いました。 恐らく、canは能力を問うので、失礼だと思ったのでは。 それ以来、私もDo you speakを使っています。

Yes, you are right. 'can you' is asking about ability, and 'do you' is asking about your habits. That's certainly the case with swimming. In terms of language, 'can you' is more direct, if it is unclear (e.g when you go to a counter in a non-English-speaking country). I don't think it is normally rude, but could be in certain circumstances it could be. e.g. I might occasionally say that to telemarketers with really bad accents! It is possibly even more polite, as 'can you' can be interpreted as both 'do you have the ability' and 'can you please'.

The difference lies in whether you do something in general or whether you're currently doing it right now. Eg: "Do you play soccer?" This is asking whether you play soccer generally. "Are you playing soccer?" This is asking whether you are currently playing right now. Eg: "Are you making supper?"(Current) or "Are you making supper right now?" "Do you make supper?" (General) or "Do you ever make supper?" I hope this makes sense :) Don't hestitate to message me if you have any questions? I'll do my best to help.

Thank you so much!

Yes. That's correct.