1. Will this affect the functioning of the firearm?
No, absolutely not
2. Is this a chip in the finishing or the Tenifer itself? I'm wondering about longevity and resistance to rust. If just finish, no issues there.
Your picture will not enlarge, so I can't tell by looking, but my instinct says that it is irrelevant. Do a search on "tenifer" to understand what it is (more a treatment than a finish). if you want, you can hit it with a spot of oil, or (more work involved) some paint.
3. Is this common? Are there others that have similar nicks/chips?
no, and yes. Chips, nick, and wear can occur anywhere.
Keep shooting and collecting marks, relax, enjoy.