I agree with what many folks are saying here.
The higher end rigs and some brands are better for fulltime and all season travel. I'd go with a Motorhome, preferably diesel for the power, ability to get Aqua Heat and size. A toad that will hold the family and dogs for outings can easily be towed by a big diesel.
I'd not travel with pets in a crate in the back of a truck or in a TT..not safe for them for many reasons.
Campground dog/pet # limit...it is hard to say. Nearly all we've been to have a 2 pet limit no matter what they are. If they are indeed in the "banned breed" category, larger than they permit, or more of them it is the the CGs opinion as to whether they will allow it or not. We've been to campgrounds that only allowed dogs smaller than 50#s. I would be honest with the CG manager at check in and not try to lie because you will have to walk them on a leash, cannot leave them outside alone and could be asked to leave if they bark a lot. They will know if you tried to hide that fact. Even if you will be fulltime for a short time you may still end up staying in a park for a couple of weeks or more so indeed you will have to just work it out with each individual CG. We traveled with a dog and 2 cats for 10 years and never had an issue but that is a lot different than 3 large dogs.
We lived in our MH fulltime in our home locations (shown in siggy) for 3 winters, had tons of snow and temps to -25 with the wicked winds our region is notorious for. We managed but we had to fill fresh water and drain tanks then stow all hoses somewhere warm...which will take up space in your heated bays. We used two residential size propane tankfuls each winter! A heated floor is the best way to go so you can refill the diesel from containers and ehat the floor better. Even with basement heated by the gas furnace when the temp drop below 10 and stay there for quite a while the floor will be very cold for humans and dogs to walk on whether carpeted or not. We put weather station sensors in the utility bay to know when we needed to add a 60W bulb...which was when the temp dropped below 10. In extreme cold an absorption refrigerator may stop working and not come back on until the weather warms up so I'd opt for an AC run residential unit.
I'm going to assume you will be going north in the summer and south in the winter so the previous info is "just in case you don't".
Some of the bunkhouse type coaches would be great with kids so you aren't constantly making your couches into beds. With a family of 4 and 3 dogs you will find when the weather is not great those 4 walls and tight quarters may get a bit hard to handle.
Having so many on board...you should learn to live in a "bare necessity of life" mode with only a few "extras", ditch all the electric kitchen appliances, keep clothing to a strict minimum (just do laundry more often), don't overstock food and keep cabinets packed loose enough you don't have to dig to get things out and repeat the process to put stuff away--been there done that and it was a complete miserable experience. Most rigs today have a lot of storage and you can indeed pack literaly a ton of stuff in them but that is just a waste and you'll find at some point you will regret it.
It sounds like quite an adventure. You will find out what will work for your family and what won't. Just be prepared to alter things as you learn. Enjoy!