I got my horse a few months ago for $1,000.
Horse prices are low right now. A lot of people sell horses before/during the winter to keep costs down.
I really don't think $1,000 should throw up red flags. Just my 2 ¢ though.
She is a cute pony. She could use a better rider, but she looks good for what she is working with. She does look green on the xc course, but if you take the time and work with her and not rush her, then she'll be good.
If she's not registered, you just can't show in any breed shows. But you can register her with other registries. (I'm going to register my gelding with the Performance Horse Registry.)