The answer to both questions is yes. A vertex-transitive connected finite graph (with more than two vertices) is $2$-connected, i.e., it can't be disconnected by removing a vertex. In a $2$-connected graph every pair of vertices lies on a cycle; this is a special case of [Menger's theorem](https://en.wikipedia.org/wiki/Menger%27s_theorem#Vertex_connectivity).