Is it correct to say "Can I know where you live?"
That would sound quite odd if you simply want to know where the person lives.
Saying "Can I ask where you live?" would be more typical.
Yankee, what about:

"May I know where do you live?"
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OR you can say:

Can you tell me where you live?
SpocikYankee, what about: "May I know where do you live?"
Hi Yankee,

I hope you don't mind my replying on your behalf.

Not possible. It should be "May I know where you live?"

You can ask "Where do you live?"
Yes, saying "Where do you live?" is the most direct form of the question.

Saying "Can I ask where you live?" is more indirect and would be appropriate if you want to ask the question a little more politely. I would use 'ask' and I would never use 'know' to find out where someone lives.

Using 'know' might possibly be used in a different context to ask a completely different sort of question.
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I'd suggest another alternative for a polite question:

Would you mind telling me where you live?
Yes, there are a variety of ways to word that question less directly and more politely, which is not to say that you can never use the simple, direct question.