This isn't a complete answer, but just an idea about your
first question. Let $P$ be a rank one projective module
over the commutative ring $A$, and $P^*=\mathrm{Hom}_A(P,A)$
be its dual. Then for $M=P\oplus P^*$, $\bigwedge^2 M\cong
P\otimes_A P^*\cong A$ is free. There must be $P$ for which $M$
isn't free, but I can't think of any off the top of my head.
If $A$ is a Dedekind domain then $M$ is free. Taking $A=C^\infty(N)$
where $N$ is a smooth manifold, then $P$ would correspond to
a line bundle on $N$. If $M$ is free then the direct sum of this
line bundle and its dual would be trivial. Surely there are manifolds
and line bundles for which this isn't true?