Yes, it should be $\wedge^i T_A$. Btw, if I remember well, there is an dg-alg. A (finitely generated over k) s.t. its Hochschild cohomology is not finitely generated over A (which does not answer the question).

I do not think it is true. However, in the case of smooth A, we have the Hochschild-Konstant-Rosenberg isomorphism, which tells that there is an isomorphism between $HH^i(A)$ and $\Omega^i_A$. In particular, $HH^i(A)$ is finitely generated.