Independent Node Degree in Undirected Graphs
Concerning the third question, triangle-free graphs have $\Delta^i(G) = \Delta(G)$. From Turán's theorem, we know that the maximum number of edges on a triangle-free graph is $\big\lfloor\frac{n}{4}\big\rfloor$, and this bound is tight for the complete bipartite graph $K_{n/2,n/2}$. Hence, at least in this case, you can have a dense graph with a large independent node degree, namely $d^i(v) = \frac{n}{2}$ for all $v$.