Both Theorems are correct. For a reference of Theorem 1, see e.g. this math.SE question, discussing the theorem that a unital ring is a division ring if and only if every unital left module is free. For the second theorem, see e.g. this math.SE question discussing precisely this question.
As for generalisations. For Theorem 1, Artin-Wedderburn Theorem can be viewed as a generalisation of it. The class of free modules is replaced by the more general class of projective modules (which can e.g. be defined as direct summands of free modules). One formulation of this is then, that every module over a ring is projective if and only if the ring is isomorphic to a direct product of finitely many matrix rings over division rings. See e.g. the wikipedia article on this theorem and the references therein.
For Theorem 2, the property you are looking for is called the invariant basis number, meaning that all finitely generated free modules have a well-defined rank. This is satisfied for any commutative ring, any left noetherian ring and any semilocal ring. But there are also counterexamples for general rings. For references, see e.g. the wikipedia article or this mathoverflow question and the references therein.