This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory.

Suppose $\mathscr{X}$ is an algebraic stack over a scheme $S$. What is the correct notion of a tangent space to a point $p \in |\mathscr{X}|$, as a tangent space to the object in the stack and as a tangent space to the set of points?

Whenever we work with the coarse moduli space of semistable sheaves $M$ on a variety $X$, we can identify the tangent space to a point $[\mathcal{E}] \in M$ with the group $\text{Ext}^1(\mathcal{E},\mathcal{E})$, but these groups come from the deformation theory of coherent sheaves. I would like to know how one can work backwards, in a sense, and go from a stack to the deformation theory giving rise to the stack without knowing if this stack represents a moduli functor. I suspect this has something to do with Artin's criteria, but, as mentioned, I am not fluent in deformation theory.

Now, suppose we have a good notion of a tangent space to a point $p \in |\mathscr{X}|$ and $\mathscr{X}$ is irreducible. If $T_p\mathscr{X}$ is trivial, then is $p$ the generic point of $\mathscr{X}$?

The question seems odd since an irreducible variety with vanishing tangent space to a point is a single point, but, again, the motivation comes from semistable sheaves. In some cases, it is more convenient to work with a larger stack of sheaves $\mathscr{X}$ which contains the stack of semistable sheaves $\mathscr{M}$, such as in here. If $\mathscr{X}$ is irreducible and $\text{Ext}^1(\mathcal{E},\mathcal{E})=0$ for some sheaf $[\mathcal{E}] \in |\mathscr{X}|$, then the expectation is that $[\mathcal{E}]$ is the generic point of $|\mathscr{X}|$, but not all of $|\mathscr{X}|$ due to the fact that $\mathscr{X}$ contains a lot more isomorphism classes of sheaves than $\mathscr{M}$. I would like to say that $\text{Ext}^1(\mathcal{E},\mathcal{E})$ is the tangent space to the stack, but I'm not sure if this is correct. For example, what role should the automorphisms and obstruction space $\text{Ext}^2(\mathcal{E},\mathcal{E})$ play in determining the tangent space?

If we rephrase this using morphisms, this is equivalent (I think) to asking whether a morphism $F: \text{Spec}(K) \to \mathscr{X}$ sending $\text{Spec}(K)$ to $p$, where $T_p\mathscr{X}$ is trivial, is a dominant map. In light of this, we can ask more generally:

Is a morphism $F: \mathscr{X} \to \mathscr{Y}$ which induces a surjection on tangent spaces $$F_p: T_p\mathscr{X} \to T_{F(p)}\mathscr{Y}$$ for some point $p$ where $\mathscr{Y}$ is irreducible a dominant map?

The hope is that, with the right definitions, these morphisms behave very much like submersions.

I hope I've made my questions clear, and I'm happy to work through some references if these are very basic questions.