As mentioned in the other comment, dogs 'do their business' while on walks, but I wouldn't call it wetting or soiling themselves since they're doing it where they're meant to do it. The dogs in the game seem to be perfectly house-trained and won't pee or poop indoors.
Though it isn't dog-fighting in the sporting sense, if you meet another dog on a walk and your dog doesn't get along well with it, they will attack each other. The fighting isn't very graphic, mostly just lunging at each other, but it could still be upsetting.
If you choose to donate your dog at the Dog Hotel, the dog you're donating is shown on-screen looking sad, then looking away from you sadly once you make the final decision to donate it.
Dog breeding does not exist in this game. This was just a fake rumour that circulated in the 2000's. In fact according to the creators, the dogs in the game are just puppies and aren't sexually mature anyway.