The river will still fill that diagonal tile, so all that breaching under it would do is flood wherever it drained to. If your river freezes in winter, you could pump out the extra water and seal it then, otherwise it's an aquaduct permanently (barring cave-in plugs or obsidian plugs, but those are beyond the scope of the question).
The dog could have been a migrant's pet when they showed up, it happens sometimes.
What you need for tameable animals is, from the Z -> animals list, a stray (animal) listed as unavailable. They cannot be in cages or restraints. When you queue the train war animal (or hunting animal) task, the animal training dwarf should go find a stray and take it to the kennel for some training, and the critter emerges as a stray war animal, unavailable. The stray will follow the dwarf that trained it until you assign war animal through the [ v ] iew menu on a dwarf. Good choices for war animals are soldiers (esp ranged), woodcutters, or miners.
The on the z -> animals screen, you can highlight a stray and press enter to flag it as available, which frees it for adoption by a dwarf as a domestic pet. This is handy if you have, say, a breeding pair of critters that you don't want to accidentally butcher.
The exception to the pet rule is cats, they refuse to be assigned to anyone and will eventually adopt a dwarf on their own, making them immune to butchering. Get 'em while they're young, catsplosions are a terrible fortress disease, ruining FPS.
Speaking of, animal population control is usually done by a butcher. The z -> animals screen lists critters by age, so the kittens are at the bottom. Alternatively, you can [ v ] iew individual creatures and manually flag them as ready for slaughter. Butchering, fat rendering, and hide tanning jobs should be created automatically, if you have a butcher's shop, kitchen, and tanner's shop built and dwarfs with the associated labors enabled. Make sure there's a refuse stockpile for the spare bits like connective tissue and feathers etc., and a leather stockpile for the tanned hides.
Frogs and the like are considered vermin, and aren't big enough to be used for cooking or butchery. If you have a dwarf who likes that animal, moving the cage to their room or office would provide happy thoughts, otherwise you can link the cage to a lever and let the animal out, or put the cage somewhere public and stuff a bunch more animals in it for a zoo display.
Industry flow and efficiency is largely managed by specific stockpiles. These can get fairly complicated and are time-intensive to reorganize, so a bit of planning goes a long way in the early years. Continuing the example above for a meat industry/food...
I try to keep my food and booze production near the dining hall for efficiency's sake. Stills have a food stockpile set to only accept brewable material, to minimize travel time between barrels produced. Butcher's shops require a nearby refuse stockpile for extra chunks, and also to dump rotten bodies that weren't dealt with fast enough. (tip: miasma doesn't spread diagonally, but dwarves can walk diagonally, so leave a pillar in the front of your refuse pile so miasma doesn't leak out and make everyone unhappy) Once an animal is butchered, it yields a pile of prepared organs, meat, raw hides and fat. All of these except the hides are processed in a kitchen into prepared meals, and tallow, so a stockpile near the kitchens that accepts only organs and meat and whatever plants products you'd like to cook (quarry bush leaves, dwarven sugar, etc) helps out there. Also, the kitchen should have a nearby output stockpile that's close to the dining hall that only accepts prepared food. The raw hides should get taken to a nearby tannery, with adjacent leather stockpile for the output, that feeds a leatherworker's shop.
That sounds like a lot, and it is, the food industry's very robust. However, you don't have to fit all of that on the same Z-level, I usually use 3 or 4 floors for my food industry, with raw materials in the middle and outflow near the top and bottom.
Military is a complex and robust system. I'd get lost if I tried to retell all of it off the top of my head, but I do need to set one up in my new fort as soon as migrants come, so I'll do a mini-walkthrough of it later on when I have the game open to reference.