1. 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.
2. 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....
1. 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.
2. 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....