Cayley subgraph isomorphism and complexity of linear subcode decision
For your first question: yes, it is NP-complete. The complete graph is the Cayley graph of any group, if you take the entire group (say, except the identity, but that would just add a self-loop at every vertex) as the generating set. Then CLIQUE is a particular instance of your problem. 

A more interesting question might be to restrict Cayley subgraph isomorphism to Cayley graphs where only a minimal, or say $O(\log|G|)$-sized, generating set is used.

For your second question, the answer is also yes, because not only is Code Equivalence similar to Graph Isomorphism, but GI reduces to Code Equivalence. Essentially the same reduction (due to Petrank and Roth) gives a reduction from Subgraph Isomorphism to Subcode Equivalence.