Using the h-principle, Gromov showed that there is a symplectic form on $\mathbb{R}^6$ which admits $S^3$ as a Lagrangian submanifold.  Using holomorphic curves, he showed that the standard symplectic form on $T^* \mathbb{R}^3$ does not admit any such Lagrangian.  There is now a whole industry of building exotic symplectic forms on non-compact manifolds (see papers of Seidel-Smith, Mark McLean, ...).

Probably the only reasonable answer to characterising cotangent bundles uses the existence of a Lagrangian foliation by planes.  If you have a foliation parametrised by a manifold which admits a Lagrangian section, then you have yourself an open subset of a cotangent bundle (this is just Weinstein's theorem).  You can't drop the condition of the existence of a section precisely because you can add the pull back a $2$-form on the base.