In my opinion this is all symptomatic of some underlying relationship/marital issues with the OP and wife.
No household can have 5+ animals unless EVERYONE is on board. Wifey as you know, there is much MUCH more to a dog than walking it to the bus stop and putting him in a dog run. MUCH more than scooping out a litterbox.
I love dogs. Our dog is a family member, a HUGE part of our life, and as a dog person I can really understand wanting a dog. But you guys already have 3 kids... 4 cats.... 3 bedrooms or 5 doesn't make much difference unless you're turning your home into a safari. This is a lot of animals by any standard.
If you end up with a problematic pup (very possible with 4 cats 3 young kids and a resentful husband) the problems will be tenfold.
Wifey, in my opinion you are being selfish. And this excuse about losing your last dog being heartbreaking to your 3 year old and 18 month old is.. A stretch. Heartbreaking to you, absolutely, but in my opinion you should stop pawning this entire thing off as "for the kids" and just admit that YOU selfishly want a dog. It's ok to be selfish, but buck up and admit you want a dog. This pup is going to have a hard enough time figuring out his leader/owner in your household as it is, last thing it needs is for you to have the mindset that it's not your dog when it very clearly is. And I really hope you aren't steering your 8 and 6 year old into "wanting a dog", as that's the impression I get with your posts.
If you all didn't have 4 cats already I would be 100% on the wifey's side. But 3 kids, 4 cats, now a dog... That's a LOT of animals and everyone needs to be on board to make it work.
Wifey: You need to accept that you can't always get what you want, when you are admittedly going to depend on your husband for help with taking care of the animals. Did he want the other 4 cats? Sounds like he didn't. So at what point are you going to be ok with having enough animals? Reading between the lines of your posts, there's an awful lot of direct emotional resentment and emotional negativity in response to your husband's logical opinions about not wanting a dog. In my opinion that needs resolved before you bring another dependent family member into your home.
Husband: Seriously, dogs aren't that bad. In my opinion the best way to handle this as man of the household might be to lay claim of the dog as yours. You research and pick out the breed, you handle training him, and you make it clear that there are no more animals in the house until you guys get down to two animals. Or else you'll end up like some of these other posters with 10+ animals that you're supporting. Nothing wrong with adoption/type stuff, but with 3 kids you need to make sure your family's priorities are straight. It won't be long until your youngest wants his own dog, then the middle kid wants a dog, next thing you know you're one of the posters in here with 10+++ animals and thousands of dollars a year dumped into associated costs.. On top of the mental stress.
You all are already in a very small minority of people who have 4 animals in the house. Through my life I've seen a number of family/friends in similar situations, and it NEVER stops at 4-5 animals. People either have 1-2-3 pets, or they seem to balloon into 10++ over just a few years.