Chromatic number of G+v where G is a cograph
I think Artem is on the right track with perfection:

As cographs are $P_4$-free, cograph+v is $C_5$-free (and $C_{2k+1}$-free and $\overline{C}_{2k+1}$-free, $k&gt;1$) and so they are perfect graphs.

This means the only thing that is pushing the chromatic number up is clique size. So if $\chi(G+v) = \chi(G) + 1$, it is because v has increased the maximum clique size, which means that $N(v)$ must have had a clique of size $\chi(G)$ and so $\langle N(v)\rangle_G$ had the same chromatic number as $G$.

So it seems the statement in question 1 is true.

But even without it, since cograph+v can't have any large induced cycles, it is contained in the set of weakly chordal graphs. A graph $G$ is a weakly chordal graph if for every cycle of length at least 5 in $G$ and $\overline{G}$, there is a chord. Weakly chordal graphs are a subset of perfect graphs and there are various efficient algorithms for a number of optimization problems on weakly chordal graphs, such as colouring.