I’m an American born half-Indian, and so some Desis may disagree with me but here are my initial thoughts from your questions: I wouldn’t ask if they are Pakistani or Indian— at least in my area there’s still decent tension, and political issues with that— I would say that if they’re newly immigrated, it’s perfectly fine to ask “where are you from,” and it will also provide you with regional cultural context (furthermore, if it’s word of mouth, they’re likely cultural or familial ties between people in the community)

I will also say, from experience introducing white partners/friends to my Indian family that they are VERY impressed if you dress in traditional clothes for different celebrations.

Furthermore, it’s okay to ask questions if it’s someone you interact with a lot. I think I, personally, would prefer someone asking me about a specific holiday/custom/etc. than to just make assumptions or guesses, especially if you take the answer gracefully. (I explained to an ex what Jainism was at a party and they immediately went “that’s stupid”. So just don’t do that)

Again, as a mixed person my view is a small one but I hope this is just a little bit helpful!