So I did a bit of testing today and found out this.
Dogs:
Dogs seem to be a lot easier to breed. I started a new game and tested this out with 3 dogs 2 females, 1 male. 1 female had all positive traits the other had all negative traits one of the big ones being aggressive. If you become good friends with your dog you can tell them to "be nice too". The only time on my dogs this option didn't seem to pop up was with a dog with the aggressive trait. Strays you can invite to your house so its a good idea to do so to give you time to raise there relationship with them. If you do choose to invite them to your house make sure there is toys and food for them as it seems they interact with the stray when eating and playing. Traits will help in this area. My dog that was friendly was fine with the fact the stray was eating from the pet bowl, but my aggressive dog didn't want the dog around any of its toys or food. If you do leave down food with an aggressive dog make sure to put some space with the food bowls. If your dog does some of the negative traits it becomes harder to influence your dog to breed. For example one of my dogs were stubborn and independent so at times they did not want to listen. Its not impossible to breed dogs with these traits or with the aggressive trait but it takes more effort. Even if your pets are growling or pawing one another when you tell them to breed there is still a small chance they will, there is just a higher chance when they nuzzle and there's hearts. If they growl that's more like a 10% hearts is more of a 35% chance. ( not too sure on the % this is just is just the rate it feels like.) When you are breeding with strays you have to remember to make them happy example feed them bath them comfort them if there sad. When I was playing if the stray was sad or there needs were too low they did not seem to want to breed. So make sure to keep an eye out for both pets needs you want to breed. Household pets; If your pets are not spayed or neutered and are always happy with needs taken care of with a good relationship with the other animals they will breed on there own. I was using cheats to cheat the needs of the household for this experiment doing so my pets already had about 2 litters of puppies each.
Cats:
cats seem really hard to breed if you don't have 2 in your household you want to breed. I did manage to breed them outside the household but it took a bit. The reason I say this is because there is no "be nice too" with cats (unless my game is glitched ) I did the same strategy with dogs only I found cats that lived in a household were a lot easier to convince my cats to breed with. I managed to get 2 of Caterina's cats pregnant with Izzy one of my male cats after a 2 visit's by telling him to breed a few times and just letting them hang out in her house. They did not show successful hearts instead they pawed each other. Not sure what it meant I still have to do more research on cats but it worked now there both pregnant.
This is all just from me testing game-play with just cats and dogs no house on the lot although I did put toys, food, a bath and beds on the lot just to see if they would get along better without me telling them to. I gave all the pets different traits. It does seem like some pets did get along better with other pets with the same and similar traits but I didn't test that too far. I know my lazy dog did not want to breed at all with an active and playful stray we found. I had to work on there relationship a lot more then others. The cats seem to hang out more with each other on other lots around the lot I played on and the more they hung out the easier it was but none of my cats had the lazy trait so Id imagine it would be different. One of my female cat, Honey got pregnant 2 times once after some influencing the 2nd time no idea she just poped out kittens from the same father of the first litter of kittens. If anyone else has something they've noticed Id love to hear it.