As a freelancer, I can tell you that if you appropriately set expectations with your clients, you can answer emails whenever you want. Personally, I answer all of my emails at 11pm. Nobody else is awake and there's no danger of an immediate reply.

If a client has an emergency, they can call me. My phone hardly ever rings.

To answer your question - no, I don't charge for emails unless the volume is excessive (> 10/day or so, not counting notifications from PM systems and the like). At that point, I just bill them an hour for "Communications and Project Management" and call it good.