My black Lab is supremely laid-back and (I really noticed this when I was taking him and the terrier cross to doggie daycare) absolutely refuses to engage in any sort of conflict. He is not at all a pansy or subservient dog - he will simply turn away from conflict of any sort. He’s not a wuss - I’ve watching correct and uppity pup who tried the dominance-humping move, but other than that he’ll do just about anything to get along.
He is completely comfortable with being the beta dog in most situations, and I’ve seen this at friends’ houses with their dogs, at doggie day care, and at a park I sometimes go to with other dog owners. He’s extremely bonded to the terrier cross; when she had to spend a few days at the ER vet last year he was utterly discombobulated.
The little terrier cross bitch and the asshole Rottweiler have worked their relationship out; neither is clearly “alpha” (I dislike that term for being too simplistic, actually) and they each have areas where they’ve agreed to let the other one be the boss.
That’s poor terminology really; I don’t think dogs have an egalitarian bone in their bodies, but if they’re well-balanced dogs, they’re good about figuring out which battles are worth fighting!
The terrier bitch not only hikes her leg and marks, she back-scratches afterwards. Many bossy bitches do this.
But, the Rottweiler always, always, marks over her. However, she doesn’t care.*
The Lab goes off in a quiet corner to squat and pee in peace. 
*Because she is calmly confident in her supremacy of My Bed. She’s the only one allowed on the furniture. She knows it, I know (and allow) it, and doG help either of the boys if they attempt to encroach on her literally elevated status.
She was raised by Rottweilers - I had three when I got her as a puppy and she was a teeny thing compared to them - so she takes her literally-elevated status seriously. And I let her.
The terrier cross owns all toys. Except the big ones, like the tractor tire and the Jolly Ball, which are bigger than her so she lets the asshole Rottweiler have them.
The asshole Rottweiler must be the first of the dogs to [del]explode[/del] go out the door to the back yard.
The Lab, however, has a giant stuffed toy dog that he carries around and sleeps on. For some reason I’ve never been able to ascertain. the asshole and the terrier, both of whom are awful Destructo-Dogs, have left it alone and defer to his ownersip of it, since April 2011 when I brought it home.
The asshole Rottweiler has an enormously high self esteem and expects everything first. He isn’t pushy about it, he just expects it. So with feeding, the order is him, then the terrier, then the Lab. It’s become such a routine I think they’d each be upset if I did it in the “wrong” order. Bottom line though, I own everything; everyone including the asshole must wait for permission before bolting out of the front door or a vehicle, and the asshole must sit and wait for permission before eating.
They all have to work for treats and rewards. I play games like “sucks to be you” with treats: eg on the sit command, the slowest to sit…well, sucks to be you, no treat. And you’d be amazed how quickly dogs catch on to this sort of thing!
So, yes - based on both tons of reading and tons of experience, I really think “alpha” is not a very useful term and analogies to wolf packs are inaccurate. Perhaps it’s a little different in a pack of wolves, or pariah dogs, where there is a: less human intervention and b: more “natural selection” when it comes to breeding, pack order and organization.
kevbo, I hope you find a buddy for PJ! Sounds like you already know what to look for - male, not huge, laid back. I know if I were to lose both my boys tomorrow, the terrier would want company (or, someone to boss around LOL) but I’d have a hard time finding a female dog she’d be OK with and she finds small dogs uninteresting. So I’d need to find her a big boy. 