Congratulations, and good on ya for adopting an adult dog!
Snapping/snarking/growling doesn’t necessarily have anything to do with dominance (less likely with a male v a bitch); he could just be feeling a bit unsure and defensive. Also if he snarked while still in his crate, even more likely it’s defensiveness/barrier frustration, something like that. (Let him know, nicely, that it’s not acceptable though!) Typically males don’t aggress on females, as long as they’re well-balanced and generally nice dogs.
I think you’re doing the right thing with the crate at first, and also your idea of taking them all for walks together is absolutely spot-on.
I have a basic routine when I bring in a foster dog - I have three resident dogs.
1: All “valuable” items are picked up. Toys, chewies, even the water bowl. Anything that I think a dog might get into their brain to get possessive about is put out of reach.
2: My dogs get kicked out, new dog gets to walk through the house and sniff things for a short while. Then into a crate, lots and lots of praise. No toy or edible items in crate in the presence of other dogs at this point.
3: My dogs come in. Once everyone is over the excitement of a new dog and are settling down going “meh”, dog gets let out of crate. Usually this takes 30-60 minutes, a long drawn-out crating of the new dog can get everyone frustrated. Depending on dog, I may do one-at-a-time introductions, or just throw them all together. NO SNARKING ALLOWED by anyone.
4: I just watch for any signs of stress, tenseness, staring or posturing and head it off at the pass. Knowing canine body language is helpful, you probably already do. I don’t allow bossing or guarding or pushieness. Usually everything goes fine - as a foster home I try to bring in dogs that are friendly and laid-back and generally will get along.
Generally, New Dog is treated as bottom of the totem-pole dog. It has strict rules (here, dogs have to earn the right to get on furniture and so on). New Dog gets fed last, whether it’s a meal or treat. And has to “work” for everything - sit, take nice, touch, whatever. (Google NILIF - this is how I’ve treated all dogs in my house for years, even before it had a label.) New Dog gets tons and tons of praise and attention for doing the right thing, gets either ignored or mildly rebuked for not conforming to basic house rules and manners.
It can take quite a while before New Dog is left alone loose in the house with the resident dogs. I’ll crate, baby-gate or otherwise separate dogs until I’m absolutely sure they have any and all issues worked out. Some grumbling is normal, but all-out fights aren’t a given. A general rule of thumb - males will have big noisy fights to establish position but there’s often little damage done. Bitches are much more likely to fight to kill, plus they tend to hold a grudge and be less forgiving than the boys.
If I seem a bit over-cautious it’s for good reason…I made a big mistake with one of my first foster dogs almost 20 years ago. I left three bitches and one dog alone in the house when I went out for a few hours, because I was sort of ignorant at the time. Came home to absolute carnage, blood,two ripped up females (the lone male, a Rottweiler, wisely stayed out of it), hugely expensive ER vet bill - the female foster dog went after my smaller female, my larger old female got into the fight…the foster was unscathed but the other two required massive stitching-up. It was horrific. And, it was all my fault. I’ve learned a lot since then and haven’t had a repeat - probably because I am now more knowledgeable and also very careful! I’d rather be over-cautious than deal with ripped-up or dead dogs.
