Unfortunately, the way you've phrased the question makes it sounds like you're asking for medical advice, so it risks being closed. Saying that though, it's also an issue I've had in the past, so I may be able to offer a little advice.
1. Critiquing the movement.
If you want to post critique videos, then you can definitely post them here. However, I'm not too sure how familiar the community here is with that particular exercise, so you may have better luck posting on a more specific kettlebell forum, such as the kettlebell sub-reddit.
2. How to "train" your toes to not hurt as much
Get stronger feet.
Now, I'm not exactly an expert on physiology, but my understanding is that there are a number of tendons that attach to the toes that run along the bottom, the arch, of the foot. If I'm understanding you correctly, the problem is likely with these than the toes themselves.
Couple of things I've found helpful, myofascial release on the foot (this is a good video, but if you search around YouTube for foot strengthening, there's a lot of good stuff all saying pretty much the same thing) and the kneeling toe stretch (ease into this one though, and do it after doing some soft tissue work on your foot, as per the first link).
I run with a modified Turkish Get Up on an almost daily basis. It's very rare that I do the full movement, though my reasons are, simply, because where I train I don't have the height to stand up with my arm overhead.
What's I'd recommend for you is to just go to the half kneeling position (bottom of the lunch position) and stop there. When I'm doing this with weight, I like to either hold that position for 10 - 15 seconds, or do a few shoulder presses while I'm there.
For most people, the weak point of the TGU isn't the lunge position, it's core strength and shoulder stability. By progressing through to the half kneeling position, and holding it, with weight, you're still getting a good amount of shoulder stability work and core activation, even without moving through the full lunge.
(Just in case you consider this "cheating" as you're not doing the full movement, these and a few other variations were given to me by an SFG certified coach as preparation for my own SFG certification).
4. Anything else
Using an empty box / shoe / bottle of water / kitten to get the movement down is all well and good, but start using weight as soon as you can. Heavy weight is instructional. You will learn a lot about your body, movement and even physics by adding weight to a TGU. It can seem scary at first, and if you've got someone who can spot for you for your first few attempts, get them to help, but definitely add weight.
Having said that, don't be in a rush to increase the weight. This is one of those movement where you really need to master the movement. Classic kettlebell training weights (poods) jump up in increments of 8kg. In my own training, I've jumped from a 16kg TGU, to a 24kg, to a 32kg. Those are massive increases, and in order to jump that much and not drop a kettlebell on your own face, you need to be extremely confident and competent with the movement. Do I recommend jumps of that size? No, I'd say go with smaller ones. Why did I do it? Because I'm stubborn and wanted to see if I could.
(This is where I deviate from someone I end up quoting way too much in my answers here. Dan John recommends against doing weighted TGU's, I think he's actually against "heavy" weighted TGU's, as the risk / reward ration doesn't seem high enough. I do understand his point of view, but I've personally found them very helpful to rehabilitate a shoulder injury, so really like them.)
Remember, the TGU is a slow movement. When I rush one, it takes me about 25-30 seconds to complete the movement. If I'm using a heavier weight, something like 45 seconds isn't unusual. Own the positions, and own the movement.