It does not sound natural in French but you should use “Oui, je le suis” where “le” would stand for French, or hungry. “Oui, c’est moi” would work well if you’re Danny Smith.

If you want something generic, it would sound more natural to say “Oui, c’est le cas” (Yes, it is the case) which works for everything.

You can just say “oui” or “non”. This isn’t like in English: a one-word answer is perfectly acceptable.

Or if you want a longer answer, you can just repeat the sentences. “Es-tu Français ? Oui, je suis Français.”, “Est-ce que tu as faim ? Non, je n’ai pas faim.”