You are right that every sitch is different and circumstances dictate a lot. In my particular sitch I moved out of the house first and I always had keys to the house and the car, but I didn't enter the house or use the car without letting her know ahead of time. Now that I have moved back in, she still has keys to the house but she has left the car keys (she took the moped.) If a spouse was being abusive about accessing the house or property I can certainly see how different arrangements would need to be made.
Regarding a joint pet from the MR I think in most cases setting up a visitation schedule (if the spouse wants it) is generally the most equitable thing to do. If the spouse has a connection with the animal it seems overly harsh to deny access. It's certainly not the same as sharing custody of a child, but I do think there is (often) more attachment than to material possessions. While you are right that it varies from sitch to sitch, I would encourage LW to consider sharing access to the dog.
W 34 Me 42 Married 7 years together 8 0 kids 1 beloved dog BD 4/6/2018 I moved out 4/7/2018 I moved back in alone 8/05/2018 I file 3/06/2019 D official 5/7/2019