As regards registration -
The breeder may have gotten temporary registration on this filly when she was born, but the registration becomes invalid the moment she sells her. If you become a member of the registry, she can transfer the registration to you, but it will only be good as long as your membership lasts (you have to pay dues to the registry every year to remain a member). When the filly turns 5 years old, you will need to get her certified as being within the height requirements of the registry, and get her registration made permanent. Otherwise, she will be unregistered.
Too many people look at a mare as a walking uterus - "hey, we can use her to make more, and we can sell them!!" The state of horses in general and minis in particular is such that you will spend more money getting a healthy foal on the ground than you will get by selling it, unless you are working with outstanding show animals. Minis frequently have problems when giving birth; getting the vet out for that can be really expensive, and you still may lose the foal, the mare, or both.
16 (feet?) x 16 is barely larger than a big horse stall - that isn't a pasture, it's a pen. As Ol Grey Mare said, it will be bare dirt in no time, so you will need hay. A lot of people only feed hay to minis, with maybe a "ration balancer" supplement to make sure that all nutritional requirements are met.
I wouldn't recommend letting your mini "play" with a Quarter Horse. Horses have pecking orders within their groups, even the usual kind of bumping and shoving that is done to sort that out could result in a dead mini.