Hi!!
If a dog does this without having neurological issues, then I'd say that he or she is feeling good.
It's a shower on their soul. When you meditate and or take a shower, you feel refreshed and anew. That's how they feel when you clean them.
As for runes. When protecting, you can use Algiz, Sowilo, Berkano to protect them with gentle energies.
If the animal is actively getting attacked by the enemy or is going through a terrible negative planetary transit, it should be good to include Thurisaz with a positive approach and a positive affirmation while using Algiz and Sowilo together, to deflect such negative things.
Just make sure to approach any animal you work on with lower reps, much like anybody new who is just starting out. You don't want to overwhelm the animal or burn it out.
Yes, you can protect and clean the animal, just visualise this clearly!!
You can refer to my post here on an Aura of protection I've used on my pets that actually worked out pretty nicely!!
I sadly had to cut out a little due to events that happened in my life where I have to get more rest now.
Anyways, here is the post: https://www.ancient-forums.com/viewtopic.php?p=436572#p436572
I hope this helps answer your questions!!
And remember, start with low reps, don't overwhelm the animal and also use positive energy when working with wild runes like Thurisaz!!
HAIL SATAN!!