"Do you mind if I ___?" is a friendly-sounding and light way of asking "Can I ___?"

Use "Do you mind if I ___?" to ask for permission when you think that the answer will probably be "yes". If you think that the person might say "no", use a different kind of question like:

Is there any way I could sleep on your couch for a few nights?

Could I sleep on your couch for a few nights?

Note that English speakers often skip “Do you” when asking a question, but in writing it should be included.