Are dogs permitted?

Yes, on leash on all but the "intermediate mountain bike trails" which are designed for mountain bikes, ie. the trails have obstacles, narrow single track, etc. Generally, if you stick to the outside trails you are good. The signs at the entrance of the parking areas have the trails marked.

On leash on hiking trails only, not MTB trails.

Idk if there's a rule about it, but with mountain bikers biking it probably wouldn't be a good idea