Any finite Graph G with all V have at least degree of 2, is it true that every vertex is necessarily contained IN a cycle?
You are asking two questions. The first question is whether your graph is a counterexample to the claim that in a graph with minimum degree 2, every vertex lies in a simple cycle.
Unfortunately, your graph is not simple, so if by graph you mean simple undirected graph, then your graph is not a counterexample. However, if you replace the self-loop by a cycle, you should get a valid counterexample.

The second question is whether the claim is true if the graph is 2-regular (all degrees are 2). Indeed, a 2-regular graph is a disjoint union of cycles. I'll leave you to figure out a proof.