I rode horses for over 25 years and worked professionally riding and breaking young horses. I rode as an exercise rider at woodbine and was used to riding basically anything and making it go better than their owners and pretty much anyone else. I can tell you it is not an easy thing to teach your horse to be comfortable to shoot from their backs.... I tried using an air rifle a few times and only one horse was "fine" with shooting from the saddle (and that was with a .223) the others where "terrified" and would do a quick deek that were you not sharp would see you on the ground fast. The one horse that you could shoot from was a retired polo pony that I was prepping for a older senior to use as a trail riding horse, part of the deal was teaching it to be comfortable with a .223 which is difficult enough to aim when standing still, I never did attempt shooting on the move but i'd imagine it to be more challenging than running on foot. The old man the horse was for liked to carry a rifle for coyote's. I'd say horseback would be a fine means of travel to a nice hunting destination but for the most part i'd leave the horses back at camp and hunt on foot. Unless you are very experienced or you are riding a bomb-proof horse that has been trained for it, i'd suggest you practise riding a lot more before attempting something that could get you pretty hurt on a lark.
Best broke old trail horses for the weekend warrior are older polo ponies. They are used to yahoo's riding them, there generally pretty docile and don't generally react to people and horses bumping all around them. Also they are a bit smaller so will be easier to get on and off of out on the trails. Many of them are trained like old school cavalry and are used to one handed riding as your other hand is occupied swinging a giant mallot. I am out of the same now unfortunately or i'd take it as a challenge to break a horse to the SKS by the end of the week.
Also for the record guns are a much cheaper hobby that horses, just a lot less women in tight pants and tall boots.
A Quarter horse is a stable breed, I would look for one that's "Bombproof". What ever you do, do not get a barrel racer, they are usually too hot. Something else to consider, you can also lease a horse, then if it doesn't work out you could return it.
Hobbles......
The Canadian would be perfect but are a rare breed, very low numbers and almost non existent in the west.
There's a shortage of good hay right now (at least in mb, and pretty sure it's across most of the prairies) so feet might be pricier than expected.
You already know that they won't be cheap, but they have more personality than any ATV out there.
I don't know about your dog situation, but growing up with Welsh ponies we had a few that tried their darndest to kill any dog that went into the pasture that they didn't like. So if you have a cute little dog be careful how you introduce your horse to it or you might have one very stomped on pup.
Another point to ponder is how close are you to a college or university? Knew of a guy close to Olds Ab who had a sizeable herd of ponies and no shortage of pretty coeds to help look after his herd. Dunno if they were looking for horse husbandry experience or were just looking to be around horses or what the reason was. It might be a way to offset costs by part boarding to a person who wants or needs to be around horses but doesn't want, or can't afford to own one on their own.
I know my niece is part boarding a horse for the exclusive right to "exercise " it once or twice a week. Kinda like part ownership of a dog where you get to take it for a walk, pick up its poo, but don't get to listen to it bark at intruders in the night.
Thoughts to ponder