Global integral model for unitary groups

For 1, the functor is representable just because it is a closed subscheme of GL(L) cut out by some set of equations. Seeing fibers as formal completions one knows that one can use the same bilinear form for defining fibers $(\underline{\text{U}}_{V})_{v}$ so it is reductive for $v\notin S$.

For 2, you see that Zariski closure is a closed subscheme of $\underline{\text{U}}_{V}$ over $\mathcal{O}_{F}^{S'}$ where both schemes are smooth and have the same dimension. So whether they are actually the same boils down to whether the larger scheme has an extra connected component missed by the smaller scheme, but this should also be observed at the generic fiber where we already know both schemes become the same.