Prove that vector space and dual space have same dimension

Your proof is correct.

By definition, a field has $0$ and $1$, and the same goes through without any difficulty (either for division rings, i.e. noncommutative fields).

Looks good to me. I believe in the case of a general field you just consider {0,1} to be the corresponding identities in the general field, which exist uniquely and are distinct among other properties as seen in the very first theorems defining fields.

If you're interested in abstract algebra and linear algebra over general fields I'd check out Artin's Algebra, which covers both very nicely and thoroughly.