Q. Can coax and cat5 be active at the same time?

A. That depends on the context of the question.

In the FIOS forums, that question usually refers to your WAN (internet) connection.
Your WAN connection between the ONT and the router can be delivered over coax (MOCA WAN) or cat5, but not both. So in that context, only one can be active.*

If you have FIOS-TV, video (except VOD) is always delivered over coax.
If you do not subscribe to FIOS-TV, no video is present on the coax.
Normal video is never delivered over cat5.

Verizon's standard install is coax from the ONT to the router.