I would like to say that the dominance theory has been well and truly thrown out of the window...even with regards to wild wolf packs.

Top wolf biologists such as Dr David Mech no longer refer to the breeding pair as alpha male and female. They now refer to them as the breeding male and female. This is because they do not fight their way to the top. They are already there. They were the two wolves that met and started their own pack and the rest are their offspring and function pretty much like a human family with regards to respect etc...
A pecking order may develop in an artificial captive wolf group where they are unrelated as somebody needs to be in charge but it does not generally apply to wild wolves. The only reason it happens at all with unrelated wolves is because they have to decide who will be the breeding pair. With domestic dogs, all bitches come into season and will breed with any male, not just one in a group mating with one male, so it would be a complete waste of time anyway.

It doesn't apply at all to pet dogs. Aggression in dogs is learnt like I said. They do it to avoid something they don't want when other options fail...and they usually do because silly humans ignore them. No dog is trying to take on any leadership role, especially not with another completely different species. And, availability of resources also dictates whether they feel the need to decide who gets what first etc...and as pet dogs are fully catered for with regards to food, water, sleeping places, toys and all the rest, they feel no psychological need to start trying to dominate each other because they figure they'll get what they need regardless anyway.
A large group of dogs was studied in a sanctuary. All of them had been in this group for at least 6 months. Their behaviour towards each other was studied down to the last detail and there was no evidence whatsoever that a pecking order existed in that group of dogs. None of it correlated up to suggest a pecking order. Packs of feral dogs are also loose with no pecking order. Pets dogs are the same.

Dominance is a word very much misused and dogs might use "dominant" behaviour in order to avoid something they don't like or want but it is only to defend resources, not to try and climb their way to leadership position. The dominance would simply be over individual situations, not a general thing.