First she needs to understand what both reins mean and want to stand still.
For the mounting issue, she's blowing you off. I'd check each step in the mounting process. Can you stand at her side? Can you place your left hand in mane and right hand on the horn? Can you lift your left leg? Etc. It doesn't matter why she's moving, just that she is. When you find the point she won't stand, get after her, lunge her a few circles ten give her the opportunity to stand, make sure you really get after her and make her move NOW. If you're riding with a single loop rein, don't put it over her head, by the time you get it over her head and send it out she won't associate the two things together. Within a few minutes she should stand.
I use a similar concept in riding, if i'm on a horse that has no interest in stopping there's no use trying to work on it. I don't focus on that until i've got the horse wanting a rest.
From there I would say "Whoa" (i'd use seat but since this isn't something desired a good stop off of "whoa' is never bad) wait a couple seconds then a one rein stop. Before I EVER pick up on two reins while my horse s moving I get a solid, soft back up. The preparation for this would be a very soft flex.
I'll pick up my reins and just take the slack out, where there is barely any pressure on the horses mouth. Wait until the horse lowers their head and or flexes at the poll and puts a bit of slack in the reins. I'll give back at that moment, give a short rub and repeat. Once the horse gets the concept of giving to both reins consistently i'll get pickier. I want the horse to not put their head lower than their withers or hide behind the bridle by over flexing. If they put their head too low, they'll be pulling on our hands and trying to dump themselves on the forehand later on. By over flexing they're avoiding the pressure. Once I have a soft give there I progress to a step backward. After the horse gives i'll keep that light pressure till I get a step back. Since the horse is just figuring it out, i'll let that softness go for a little bit then put being soft in the bridle and backing together. From there I get pickier about that step backward, I want to feel my horse lifting through the shoulders and have their hocks up underneath themselves when they back.
This will take a few days to get a decent back up where you can move to stopping. If at any point they want to move around during the backing lesson, that's fine, I lope the horse around some then offer that stop again. Don't worry about even more than one step when you're teaching this, one perfect step is much better than even two good steps. Only when the first step is perfect should you ask for two.
Once I get a soft backup, a soft stop is easy. I start at a walk and just take the slack out and wait until the horse stops and backs up a step. If the back up is good the first stop should happen within a few steps. If you have realized you don't have a good backup, point your horse at a fence for her to find that stop. From the walk, progress to the trot then the lope.
To prove a point with my last project horse I got an excellent backup, where I could just lift up my reins and she'd back softly all without ever taking the slack out. She got to this point in about three days but she's smart and I have excellent timing. To prove a point about the backup being the place to teach what two reins are about I loped her down the arena toward her pasture and picked up on my reins. She melted back down to me and stopped, just as soft as could be within a stride. That was the first time I had ever picked up on two reins while she was moving.
The lunging problem is because I suspect you don't have an excellent space bubble established.