1. Can I say it in this way？“Do you mind my sitting here?" Yes, you can. This is correct. 2. or I should say" Do you mind me sitting here?" A very interesting question! The 'grammar nerd' answer is 'no' . People who follow the grammar of English strictly will tell you that 'me' is incorrect, and that the only correct form is 'my sitting'. 'Do you mind..' is followed by a gerund form, and the gerund form requires a possessive rather than an object pronoun. The simple everyday answer is - it doesn't matter, you can say either. Most people, including native speakers, say 'Do you mind me sitting here?'. It has become an accepted phrase, and nobody is likely to tell you that this is wrong. 3. Is it "Do you mind if I sit here ?" the most popular way to say so? Yes, it is. It's shorter and simpler, and it's what you hear the most. 'Do you mind if I open the window?' 'Do you mind if I close the door?' This is what most people say. Can I also say it in this way "If you don't mind, can I sit here?" It's grammatically correct, but it isn't something which we would say. I would stick with saying 'Do you mind if I...'

"Do you mind my sitting here?" is the correct form, although you will also here "me" instead of 'my". "Do you mind if I sit here?" is also completely acceptable and probably most common. Your third option is also used, although to be more polite you should replace "can" with "may."