Yes, $S \to \Sigma$ can fail to be an immersion. The failure is a branch point of the map $S \to \Sigma$, since it is (close to) a holomorphic map. This is fairly common as soon as the multiplicity of some region in $\Sigma$ gets higher than $1$. Of course, in the simple examples you can actually compute, this tends not to happen.
A "slit" along the $\alpha$ or $\beta$ curves looks like a perfectly ordinary piece of the boundary upstairs in $S$, including at the end of the slit. It's best to think about the end of the slit as a boundary branch point, looking locally like the map $z \to z^2$ restricted to the upper half-plane in the domain. Slits cannot collide, due to boundary monotonicity: above each point on $\partial \mathbb{D}^2$, the $g$ different points map to distinct $\alpha$-curves. I don't know what you mean by "independent" degenerations, sorry....
Yes, higher genus images can happen, and it's not too hard to construct examples that are forced by gluing, although again in most cases where you're able to compute Heegaard Floer homology by directly counting curves it does not. The index formula in Corollary 4.3 of Lipshitz's paper "A cylindrical reformulation of Heegaard Floer homology" will let you easily construct examples of high genus surface with index 1, and a little more playing around should let you see that some of these must actually have representatives.
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