Should rashi and nakshatra go in the biodata?

Yes, if your side or the other side does kundli matching. These two lines let them run a quick check without asking for your full horoscope.

Rashi is your moon sign and nakshatra is your birth star. Both come from your janam kundli, not from the sun sign you read in a western horoscope or magazine. If you know the pada (charan), add it: Nakshatra: Rohini (2nd pada).

If nobody involved does matching, skip both and use that space for something people will actually read like career or personal interests.