Im the same height as you and I ride a variety of breeds. I think it all comes down to personal preference.
5'1 is a great height, I can get on most ponies and not be too big, and I have ridden horses up to 17.2 (I did look small of course :P) Right now Im riding a morgan pony (13.3 hands) a thoroughbred gelding (15ish) and a paint that's about 15.2.
Ahh yeah, qh are really cool horses too, I am sure I can find the right size for me. And Icelandic horses are adorable!! I am so glad I asked this question because I am learning so much about breed I didn't know before.
Lol I saw this tb DAPPLE horse on horsetopia I was like going to buy it cus it was REALLY cheap like 300 because they wanted to get rid of it because they got a divorce so yea but I came to show my mom and right then it got sold I was like lol