I don't have a complete answer, but I think both problems are open.
The paper by Jajcay, Malnič, Marušič  is related to your first question.
They provide some tools to test vertex-transitivity. They say in the introduction that:
For a given finite graph $\Gamma$,
it is decidedly hard to determine whether $\Gamma$ is vertex-transitive,
and the ultimate answer comes usually only after a substantial part of
the full automorphism group of $\Gamma$ has been determined.
Note that vertex-transitivity test can be done by testing graph isomorphism $n−1$ times.
Make two copies $G$ and $G'$ of your graph,
with special anchors (like paths of length $n+1$) at $u \in V(G)$ and $v \in V(G')$.
There is an isomorphism between $G$ and $G'$ if and only if
the original graph has an automorphism mapping $u$ to $v$.
Thus you can test vertex-tansitivity by fixing a vertex $x$,
and checking that there are automorphisms mapping $x$ to all the other vertices.
Also note that if vertex-transitivity test can be done in polynomial time,
then so is isomorphism test for vertex-transitive graphs.
This is because two vertex-transitive graphs are isomorphic if and only if
their disjoint union is vertex-transitive.
I believe that complexity of graph isomorphism for vertex-transitive graphs is not known.
For the 2nd question, I found a partial result.
A circulant graph is a Cayley graph on a cyclic group.
Evdokimov and Ponomarenko  show that recognition of
circulant graphs can be done in polynomial time.
Also the book chapter by Alspach [1, Chapter 6: Cayley graphs, Section 6.2: Recognition]
would be interesting for you, although it says that:
We shall ignore the computational problem of recognizing whether an
arbitrary graph is a Cayley graph. Instead, we always assume that
Cayley graphs have been described in terms of the groups on which they
are built, together with the connection sets. For most problems this
is not a drawback.
- Beineke, Wilson, Cameron. Topics in Algebraic Graph Theory. Cambridge University Press, 2004.
- Evdokimov, Ponomarenko. Circulant graphs: Recognizing and isomorphism testing in polynomial time. St. Petersburg Math. J. 15 (2004) 813-835. doi:10.1090/S1061-0022-04-00833-7
- Jajcay, Malnič, Marušič. On the number of closed walks in vertex-transitive graphs. Discrete Math. 307 (2007) 484-493. doi:10.1016/j.disc.2005.09.039