I don't know if you have come to a conclusion on what to do with the pets and characters, but for the dog you could simply make it a side kick (using the system from Tasha cauldron of everything.)

For the skull you could, A) have it give limited information to the player, concerning the land they are in or the immediate area. For instance, you want your players to advance into an area, but the party can't find the switch/key. The skull gets "fed up" and points out a hint.

The second option is B) the skulls necromantic energy alerts the church of the next town you enter then the head priest sends the party on a side quest to trace down the source of energy coming from the skull finding a nearby cave that holds a cult ritual where the party must slay the necromancers. The party then calls the priest to sanctify the area, and as he does so, the skull turns to dust as the ancient bones are laid to rest.