[Buster Brown]

Hello and Welcome to YT. Friday is such a cutie Pie. Buster was always so excited to meet everyone when he was a puppy sometimes he was a little too boisterous and overwhelmed the dogs and people he met but as he got older he learned better doggie manners and calmed down. Also be careful when approaching other dogs because Friday has not learned to tell the difference between a friendly dog and one that might bite. I learned this the hard way. I would always ask people is your dog friendly this neighbor said oh yes and then the little Chihuahua bit Buster on the nose and the owner went well she does do that sometimes (REALLY!) so you have to be careful. I was lucky it was not A BIGGER DOG!
Now in regards to potty training patience is the key and accept mistakes because mistakes mean that you as the owner were not properly overseeing the potty training (I know that one was hard for me too!) A lot of dogs are shy and to add to that your getting upset may have scared the dog so that she does that to hide her mistakes. See you were smart enough to think of that yourself too. Maybe you could put a screen up or a plant to give her a little privacy.
You do have to be careful when socializing puppies with other dogs or going to parks or pet stores when they have not had ALL their shots because germs can be passed by poop on grass, or that is on dogs paws that they track into stores etc.
I would consider taking puppy training classes once she has her shots or a home training session. Buster had some health issues when I first got him so he did not get all of his shots right away so I had a trainer come to my place. It also helped that I could address issues that occurred in my house. Just remember even if they learn a trick with every new environment means you have to reinforce the trick. Example Buster was great at learning to sit on carpet but when he was on the kitchen tiles he would not sit. I learned that to him that was different so once I retaught him the sit trick on tile he was good. The same applies to inside vs outside.
As for breeding your dog that is a tough one. Many people think breeding is easy but I have seen many threads where tragic outcomes happened especially with inexperienced breeders. If you take some time to read about all that it takes to correctly breed and deliver it will overwhelm you with all the time and hard work it requires. Please do some research and I think you will agree. If you do breed her you will have to make sure that she and the other dog do not have genetic issues that you as an owner would be responsible for as part of your contract with anyone who buys a puppy. Many states have Puppy Lemon laws California does. I bought Buster and he had several health issues when I got him. Not only did I have to go through over 2 months of fear as Buster fought for his life I spent over $2000.00 on medical expenses. According to California lemon laws the people I got Buster from were liable for his medical bills for 150% of what I paid for him are you prepared to do that for each of the puppies Friday gives birth to? Or even worse the possibility of losing her during the births? Honestly I think it is better to spay her, continue to be a wonderful Daddy and enjoy your sweet baby girl. I wish you all the best Yorkies are just wonderful and she will brighten your life I know Buster has mine!