Yes they do.

Dogs must be leashed and are allowed on paved and turf trails at Baker Park Reserve, and are permitted at campsites at Baker Campground provided they don't disturb neighbors. However, dogs are not allowed in the camper cabins unless it's a trained service animal. The park is quite heavily patrolled by park staff and police who drive around the campground at all hours.

Yes. Standard rules apply...leashed and clean up after. We had our dog with us when we camped there.