can we have a chat?

Hi [name]. Howzit goin'? // How's life treatin' you? // How're things goin'? // [Ask about his classes, interests, ... .] // ...

Thank you~

How about in a room when the student is using a computer engrossed in his work? Will a native speaker say something like this?:

Hello, I hope I'm not interrupting; can I ask you a few questions?

That's definitely possible. Depending on how well you know each other. 'could' is more deferential, more polite, less assuming than 'can'.

Thank you~

I'd thought INTERRUPTING too formal a word; is it so? I guess more often native speakers will use phrasal verbs so express the meaning?

If it's a conversation, we can use CUT IN etc, but now he's using a computer; CUT IN seems unlikely....

When we want to be polite, we tend to the more formal. We also get wordier, adding layers to polite requests to make them more polite/more deferential/softer/... .

Can you ... ?

Could you ... ?

I was wondering if I could get you to ... .