Formal neighborhood of stable curves

Question

For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “modification at a point”, i.e. removing a formal disk $D$ around $x$, then gluing it along an automorphism of the punctured formal disk $D^{\times}$. Using that, one can describe the formal neighborhood of $X$ in the moduli stack $\mathcal{M}_g$, which I’ll denote by $\operatorname{Def}X$, as the double quotient $$\text{Aut}_XD^\times \backslash \operatorname{Aut}D^\times/\operatorname{Aut}D$$ where $\operatorname{Aut}_XD^{\times}$ is the subgroup of automorphisms that can be extended to the punctured curve. This is for example in chapter 17 of Frenlel, Ben Zvi’s book Vertex Algebras and Algebraic Curves.

I’m trying to understand what happens when we take $X$ to be a stable curve.

1. First, every node is locally of the form $V = \operatorname{Spec}\mathbb{C}[[x,y,]]/(xy)$, and every formal deformation of the node is locally of the form $V_n = \operatorname{Spec}\mathbb{C}[[x,y,t]]/(xy=t^n)$. Does that mean the deformation space $\operatorname{Def}V$ is a disjoint union over $V_n$?
2. Taking $U \subset X$ an affine open containing the nodes, we have a surjective pullback map $\operatorname{Def}X \to \operatorname{Def}U$. Say we only have a single node. Is $\operatorname{Def}U \simeq \operatorname{Def}V$?
3. If we let $\tilde{X} \to X$ be the normalization and $D \subset \tilde{X}$ the preimage of the nodes, then (I believe) there’s a map $\operatorname{Def}_D\tilde{X} \to \operatorname{Def}X$ from the space of deformations preserving $D$. Is it possible to describe $\operatorname{Def}X$ in terms of $\operatorname{Def}U$ and $\operatorname{Def}_D\tilde{X}$?

Edit: for first order deformations we have a short exact sequence $$0 \to H^1(\tilde{X}, T_{\tilde{X}}(-D)) \to \operatorname{Ext}^1(\Omega_X,\mathcal{O}_X) \to \bigoplus_p \operatorname{Ext}^1 (\hat{\Omega}_{X,p},\hat{\mathcal{O}}_{X,p}) \to 0$$ where the last sum is over all nodes. I'd like to know a similar description for the entire deformation space with its groupoid structure.