Really? Do you have a counterexample in mind? What if the norms are induced by an inner product?

Since the set of embeddings is open in the set of all maps of a certain regularity from $\Sigma$ to $M$, it has a natural Banach manifold structure.

In fact, I am more interested in the local picture (second part of the question). In this case, I am thinking of the $C^k$ topology for some $k \geq 2$. I’m also unsure about what is the appropriate topology in the space of immersions…

The second variation in the present case depends on $\alpha$ too, right?