If I deal damage with a Guard Dog to a Criminal enemy, can I advance ?

The answer is yes, because the Guard Dog is a card your investigator controls, and thus "you" are doing the damage. However, let's talk about something a bit more interesting, which is just exactly how tough your situation is to pull off in the first place.

All Criminals under Act 1a are Aloof, and thus won't engage with you. You would have to take the engage action to pull one to you and then attack them yourself -- doing damage before your dog had a chance to do so.

Or you could engage one of them and then not attack -- walking to another room would trigger an Attack of Opportunity which would let your Dog get a bite in.

Finally, the easiest way to do this is to have your Guard Dog at your side, and then successfully gain a clue in the same room as the Pit Boss. He'll immediately swing at you, hitting your dog for a counter-bite, and you also get a clue or two for your troubles. That sounds like the best return on investment for your troubles.

If you draw into a criminal from the encounter deck, does it engage or just hang out there?

Aloof enemies don't engage when they come out of the deck. This agenda makes Criminals aloof. So it would seem they just hang out in the location you were, when they were drawn.

I could see an argument that they don't actually become aloof until they are already in play and engaged with you. But that would make the whole criminal aloof thing almost completely ineffective and since we can assume an intent of having a meaning effect, I think it has to be played that they are aloof in time to avoid initial engagement.