Personally I think the proof requirement is reasonable - while you may not normally think of carrying that, without a proof requirement they can't cleanly differentiate between someone who says they have a cat allergy when they just don't like cats and/or have a really minor reaction vs someone who has a proper allergy.

On nuts - there isn't a legitimate need for someone to eat nuts on a flight, so if someone has an allergy, not consuming nuts is an extremely easy accommodation. It's not so easy to deal with a cat.