Yes, I love your horse, but not a fan of the rider. I know it's not you, but even in that later video you posted, she is STILL holding her mouth throughout the entire jump. With a greenie, it is better to have the rider overjump than underjump (as this rider is.) Remember - Green + Green = Black and Blue!
Now, I wouldn't call this rider green, but I would say she doesn't have the hands to be starting anything yet. Not to say she never will, but for now, I will say get a new rider. Just tell this young lady that you want to get a number of different people on her, just so that Tia gets the experiance. Make it sound like it's not the girl's fault she got kicked off the horse. She looks fairly young, and has plenty of time to learn those lessons later.
On the other hand, I love your mare! Tia is genuially trying to please, and to do well. I love the way she moves - very flowy and pretty. The judges will like her as she moves into the show pen ;)
About her leads - I think she is trying to get them right. She is just doing whatever she feels is right for herself. Notice going into the turns, she switches to the correct lead if she was wrong before. This is an awesome sign. I much perfer a horse to do this than let him/herself be off balance going around a turn.
I would say what someone has said before - if she lands on the right lead, turn right, etc. but I would do something slightly different - Alternate! We can worry about leads later,(as long as on the flat, you can specifically ask and recieve a certain lead) especially because she already seems to kind of know them. I would turn left, turn right, turn left, turn right, etc. This will keep her thinking, and not leaning to one direction over the jump.
Regarding rushing the jumps - once you get into a bigger arena, this will be easier. But anyway - Try not to focus on JUMPING so much. If I were riding Tia, I would be doing lots of circles, serpintines, transitions, changes of direction, etc. to keep her interested, thinking, and calm. If a horse begins to think, 'Oh my gawsh, we are just going to jumpjumpjumpjumpjump,' they will just ... charge. Like Tia is thinking of doing. Think of it this way - in your jumping sessions, just do flatwork with speedbumps. Do a circle, change of direction, serpintine, jump, circle, etc. Change that 'jumpjumpjump' mindset to the 'ride along with my rider and enjoy life,' mindset.
Watching recent vid again ...
Ugh, the way that rider just pulls her around the turns ... all mouth, no leg ... so in her face ... >:/ I'd pull her off ... now...
COOKIES to you for reading that massive novel! Hopefully something good came out of it! I wish I was your neighbor - I would ride your sweet pony for you :3