from the short video you posted, it looks like she hurts but it also looks like you are kind of unbalanced and all over the place and pulling on her mouth a bit. I know it wasn't a riding critique video and I'm not trying to be harsh. I think it will be easier to ride her once you get her settled a bit more and understanding what you want. I couldn't post that trot (if you could call it that) either. I see what you mean about trying to break into a canter but it looks like it is because you are giving her conflicting aids, not because she's super sassy. I only say these things because they are things I sometimes do myself- not because I'm trying to criticize your riding.
As far as an ex racer, I have one too and he had all of the issues mentioned above and more. We have come a long way in a short time due to lots and lots of ground work and consistancy on my part. And then more ground work. When it comes to girthing, he used to try and bite- TB's have very thin skin and they aren't slow and gentle at the track so they learn girthing sucks in general. I started out by making sure he'd let me touch his girth area, then brush it, then girth him. Everytime he bit I threw my elbow up so he got it in the face. That way, it isn't like I'm smacking him to make him head shy, it just seems like he made a bad decision that didn't work out so good. Always do up the girth slowly and in stages, as mentioned above. I always saved the treats for when he let me girth him without biting though, I don't like the idea of giving him one to take his mind off of it, to each his own.
Your gate issues will also get better with groundwork, assuming it isn't a pain problem for her. Once she is respecting you and listening to you on the ground, it will help transfer to the saddle. One last tip- be careful using a whip while mounted. I'm sure I'll get flamed for saying so but my boy and a few other OTTBs at the barn get really indignant when you use it to ask them to go faster. Now, I use a lunge whip, I carry a dressage whip when I ride and can sort of slide it on his rump or around his girth or on the shoulder, but if I even so much as flick it at his hind end he bucks endlessly. I have ridden out several said bucks, acted like nothing happened, continued asking him to do what I wanted and didn't stop until he complied. But every time I try using it to get him to listen, he acts like he is mortally wounded. I get better results asking firmly with my legs, seat and voice. But you'll be able to feel your girl out and see how she does.
TBs are awesome once you win them over, but they can be choosey with who they give their hearts to. Patience, and firm, fair leadership will help you get there with her. It looks like you have a trainer which is also the biggest tool you can have in your box. Good luck with her and please keep us updated. Congrats on your new girl!