I don't believe that this is correct. The easiest way to see this is to look at your second question: The automorphisms/deformations/obstructions of a curve come from $H^i(C, \mathcal{O}_C)$, T_C)$, i.e. they are the sheaves
$R^i p_*\mathcal{O}_U$p_*\omega_{U/\overline{\mathcal{M}_{g,n}}}^\vee$
where $p : U \to \overline{\mathcal{M}_{g,n}}$ is the universal family, and $\omega_{U/\overline{\mathcal{M}_{g,n}}}$ the relative dualizing sheaf. But these do not depend on $\overline{\mathcal{M}_{g,n}}(X, \beta)$ !
In the end, I think the issue is that you have the wrong exact sequence. What you want (to produce the relative obstruction theory) is the complex
$R^i p_*f^*T_X$
where the maps $p, f$ arise in the universal diagram
$\overline{\mathcal{M}_{g,n}}(X, \beta) \longleftarrow_p U \longrightarrow_f X$
It is not obvious to me that your sheaves should be the same as these ones.
