For your first question:  The condition for integrability of $\mathrm{ker}\alpha$ when $\alpha$ is decomposable is that $\mathrm{d}\alpha = \lambda\wedge \alpha$ for some $1$-form $\lambda$.

For your second question:  The answer is 'no' in general.  If this can be done, then the tangent bundle of $M$ can be written as the direct sum of $n$ oriented $2$-plane bundles, and it is easy to give examples of manifolds for which such a decomposition cannot hold.  For example, just take $\mathbb{CP}^2$ with its standard symplectic structure.  Its tangent bundle is not the sum of two line bundles of this kind for reasons having to do with its Chern and Euler classes.