Okay, watching the video now.
The whole trotting on the line thing. Meh. She knows that you're just going to slow down and not make her do it, so she doesn't even bother trying. If you really want her to trot on the line (which isn't a huge deal, in my book), MAKE her. Carry your 5' lunge whip at your left side pointing back and down with the lash tucked up in your hand that's carrying the whip. Start to jog. If she doesn't break into a trot, flick the end of that whip over behind your body and tap her hard on the flank. Do this until she starts trotting with you, then settle back to a walk after a few steps of trotting. Rinse and repeat.
Get those shoulders up and back. Walk like you're confident and in charge. Don't look at her. You lead. She follows. If she doesn't follow, she'll get a sharp yank on the halter to remind her to keep up.
When you're yielding her hindquarters, she's doing it on her terms. You also waited too long to correct her when she put her head down to sniff. The SECOND she even tries to put her head down, you bring it back up. You're body language when you're making her yield is very passive. Shoulders hunched, looking at her head, angled away from her body. Full face her hindquarters, shoulders up and back, and CONCENTRATE on what part you want to move. At 5:35-6 and 5:46-9, what you did was almost perfect! You notice how she moved her hindquarters away without you having to touch her? Because you were looking AT her hindquarters, not her head. At 6:24-38, you're not looking at what part you want to move. You're looking at her head or her barrel, which is confusing her. That's why she's not moving. I can go out, at liberty in the turnout, and yield any part of my gelding's body just by looking at it. You stare at it and think "Move!" That translates to your body language and the horse gets the message.
When you're yielding her forequarters, you're concentrating on the wrong part of the body. It's her shoulders, not her head that you need to pay attention to. Notice how she wants to just turn her head away from you and/or basically walks away from you, not take that step over that you want? That's because you're concentrating on her head and thereby asking her to move head. Concentrate on the point of her shoulder and think "Move!" If she doesn't move it, poke her or put pressure physically on the spot you're concentrating on. The second she gives you even ONE step over, release all pressure.
I am REALLY confused at what happened at 9:25-56! And so is your poor mare!! I saw no infraction (and I watched it literally THREE TIMES) and all of a sudden you're spinning around in her face and chasing her. What did she do? And then when you're lunging her, you need to keep the pressure on if you want her moving faster. You can't put some pressure on, then let her relax, then put the pressure back on, then let her relax, etc. Either pressure for two or three circuits and quit or no pressure and let her just circle you a few times. Either way, you HAVE to be consistent and fair. You're lack of consistency and fairness are what is causing her attitude there. She's saying "Why should I try if it's not going to be good enough?"
Overall, she's VERY good on the lead. WAY better than I expected.
I see a pretty little mare who is trying, but because of a lack of consistency and fairness, shuts down at points or gets distracted. She seems VERY sweet and would absolutely flourish with a fair, consistent hand. She looks like she would do well with showmanship, really, with how she stays right with you and pays attention when you're weaving those poles. Maybe that's something you could look into. Showmanship or in-hand trail.
ETA: The part you mentioned about trying to get her to trot, she went to bite, and you smacked her. I saw none of that. I don't have audio on my work computer, so maybe I missed your verbal cue. However, it just looked like you were starting to walk and then all of a sudden flew off the handle at her. Still, consistent and fair are the way to go, regardless of the amount of space you have. If you can't lunge her at a consistent gait, don't lunge her. Back her up or make her yield.
Do not tell me I can't...because I will show
you that I can.