Is the dog just a funny gimmick for a player - but by no means meant to give the player a lot of extra power? Or is the dog an NPC which is supposed to become a member of the party, therefore also level up together with the players and remain by their side for the entire campaign? (Or something else)?
I got the feeling that your dog is in between these options. I noticed that you thought of a system to "level up" the dog. That is an excellent idea, and your suggestions seem pretty good from a roleplay point of view. However, the total hitpoints and damage that the dog can gain seems rather limited (each "trick" can be taught only once). You risk that the dog will overpower a player early on, but lose its relevance at higher levels.
My advice - although a bit drastic - would be to split the dog's "tricks" into roleplay and combat. Then you can optionally choose to include combat tricks for the dog. However, if chosen to be included, the combat-dog should level up so that it is still useful once players hit level 10-15 or even 20. That dog will need bigger buffs... And the DM should realize that the size of the party increased by 1.
