Yes, I think $\Omega$ is closed. I will add a proof later. Yes, as a mapping $T\mathcal N \to T^*\mathcal N$ it is injective, but it can never be surjective, since $T_n\mathcal N$ is a Frechet space, whereas its dual $T^*_u\mathcal N$ is a DF-space (generalized functions of distributions) which can never be isomorphic to a Frechet space. So $\Omega$ is a weak symplectic structure. See section 48 (called: Weak Symplectic Manifolds) of here, where 48.2 and 48.8 have to be corrected as described in the Errata.

Yes, I think $\Omega$ is closed. I will add a proof later. Yes, as a mapping $T\mathcal N \to T^*\mathcal N$ it is injective, but it can never be surjective, since $T_n\mathcal N$ is a Frechet space, whereas its dual $T^*_u\mathcal N$ is a DF-space (generalized functions of distributions) which can never be isomorphic to a Frechet space. So $\Omega$ is a weak symplectic structure. See section 48 (called: Weak Symplectic Manifolds) of here, where 48.2 and 48.8 have to be corrected as described in the Errata.

Edit:

Answering your comment: It is not necessary to work with Sobolev completions. I your case you can do it. If the structure group is a diffeomorphism group, you loose smoothness of the action.

You can work with the image under $\Omega$ of $T\mathcal N$ as "symplectic dual". See 2.5 of this paper for an example of symplectic reduction, which in this case is equivalent to constructing a Riemannian submersion. Also this paper might be of interest.