@Lilius; if you think she isn't established in the basics, it always pays to go back and keep working on them. They are the foundation for literally everything else. Even as you progress up the levels, it always comes back down to the same basics- rhythm and relaxation. Once you've got those, everything else should come and if you step up and loose some of the relaxations/rhythm then you go back and establish it before moving on. Its like constant thing of two steps forward, one step back but it really does help in the long run.
I only asked about the hands thing because it was a bit confusing for me to imagine. I'd expect, when you raise your hands to back her up, you are actually causing the bit to act on her lips and stopping her from 'holding' the bit in her mouth and supporting it with her tongue. This would cause her to lift her head and hollow her back, which isn't want you want. It might be confusing her if she's had some prior training too, most horses are taught to back with just a slight increase in rein pressure, but not moving the hands up or down while asking. It is an interesting concept. Whatever you end up picking, you just need to be super consistent and clear with your aids. Make it obvious that this is for back, even if you have to over-exaggerate for a little while (eg. leaning right forward, not the rein pressure). You can always make the aids more subtle and it is no use starting subtle if the horse ends up confused.
Also another thing that might be contributing to all your problems is inconsistency in the contact. I know you say you ride with loose reins, but if we're talking dressage, collection and contact, you can't start with 'loose reins', that's the end result
I'm not saying you need to wack side reins on her or ride her up into a really strong contact or anything like that, just that she might need a little more support than what you're currently giving her at the moment. Like if you feel she is on the forehand/falling in/not flexing ect. pick up a tiny amount of contact so you can just feel the corners of her mouth. When she balances herself and goes nicely again, you can give that contact away. Hopefully, she'll start lengthening her neck and opening up through the gullet while staying over her back. This would help her flexibility enormously; a lot of the 'draft' type horses can get very jammed up in the gullet since they've usually got short thick necks. She needs to telescope her neck and carry the weight using the muscles over her top line, but you've got to show her how to 'get' there first. Its a very difficult concept to try and explain >.<
If you don't want to carry a whip while riding, you can always teach her voice cues
Like start on the ground, put pressure where your leg would be and say 'over' or whatever word you want to use. Once she knows it on the ground, you can use the same thing under saddle- if you put leg on and she doesn't listen, you can say 'over' as a verbal cue to remind her. In that way, it kinda acts a little bit like the whip, it catches her attention. Verbals cues are actually super super helpful for all things dressage and I highly reccomend them! You could even teach her something like 'steady' for when she goes to get too quick, like a verbal half halt. I find a lot of horses respond really well to verbal cues as well
With the changes of pace, it certainly pays to teach her these as soon as you can but as I said before, its no use if she isn't established in her rhythm yet. The idea behind changes within a gait is that they lengthen or shorten their stride, without changing their rhythm and eventually all of this can be done off your seat. For the first few times, you'll probably have to chase her forwards a bit, but if she is set in her rhythm, she'll realise she doesn't need to go any faster, just longer. Same with the slow, she'll probably break gait and walk but once it clicks for her, you shouldn't have too much trouble
If you are worried about her canter, just do small amounts often rather than large amounts sometimes. Something I've found really helpful is to canter half a circle and then trot the other half. Once you've done it a few times, the horse's usually start to anticipate and will get ready for each transition on their own because it is a predictable exercise. The canter portions are short and she'll be thinking 'trot' almost as soon as she's completed a few strides (depending on your circle size) which should stop her from getting fast and just running on. Dressage is really about communicating complex ideas to the horse in a manner they can understand and letting them figure out what it is they need to do. Which is why you must always reward the try and not get cross when they offer the wrong answer. They've got to have the mental 'space' to work in when you ask something new or you'll just end up with a horse that panics when they don't think they've answered correctly.
Overral, I think you are on the right track from what you've said so don't be discouraged! You must remember, you don't 'start' with the 'end' product. Like my comment about contact earlier, sometimes what you begin with is not even close to what you'll finish with, but it might be necessary steps to take along the way so the horse understands and learns
Dressage is as much a mental exercise as it is a physical one for both horse and rider, so if something really isn't working, don't be afraid to take a break and find a different way of asking. Take your time and enjoy the journey
If you ever get the chance, I'd love to see some video of your mare!