Welcome to the forum !!
I have a double bred Doc Bar gelding with Cutter Bill on the bottom side and he is a fabulous horse (19 years old now !). He's my old reliable and I can put a 3 year old kid on him and know he will take care of them. The Doc Bar line can be a little stubborn, but once they learn something they tend to remember it for the rest of their life.
Regarding your question about breeding, what disciplines are you interested in? What traits about your mare do you like and what traits would you like to breed out of an off-spring? That will help us provide suggestions of a potential blood line to cross her with.
Well your mare is VERY closely related to mine. Her grand sire is Dry Docs Cutter. I found that my horse is a little stubborn but is very friendly and good with people, especially kids. I put my 4 year old cousin on her for a quick ride and she was very calm and slow. She's more running bred on her dam's side though so she's fast.
As for disciplines, I would need a solid bigger horse that could handle ranch work, as well as some run to be a heading horse. I'm also interested in working cow horse, so that's quite a variety :P
I don't trust Peanut with kids unless we are in a corral or arena. Once you're out of the arena, sometimes I have problems holding her up.
She looks solid enough to handle ranch work, cow work comes along with that. My mare is about 15.2 HH and according to my uncle she's the perfect size to be a header. You would need a little bit of run to keep up with the calf but not too much otherwise you might pass the calf all together.