I’ve found that heat plates are problematic for quail chicks because they’re so small and get cold so quickly. I’m wondering if the leg problems might have something to do with them falling asleep pressing against the plate and leaning to one side with the leg out for support. The way that one is standing tall to touch its back to the plate is not great, the plate should be high enough that it touches their back when standing normally, and they don’t have to stretch for it, they can’t sleep and stretch upwards at the same time. I realize that chick is at the tall part, but they’re stupid and it probably has no idea it’s easier to reach further in. I would put the tall side at natural back height and the short side at slightly above back height when laying down. But honestly I would use heat lights for at least the first week, if you have weaklings, longer. I started with a heat plate, for my first hatch, and I soon switched to bulbs because it’s easier for the chicks, plus you can watch them more easily to see any issues.
I feel pretty confident that the heat plate is causing the issues, after seeing that chick stretching upward, if they’re spending a lot of time tip toed, leaning forward, with the plate balancing them, they aren’t training the proper muscles for standing and running normally, so when they’re out from under the plate, nothing is working properly and they probably try to tiptoe, and end up off balance from not having the plate to push against, and topple over, so then they stick their legs out at weird angles for balance and train them into the wrong position. I’ve had chicks with this issue when they were struggling on their side for a couple days after hatch and then when they were stronger, they missed the vital period of time when they would train their legs to bend and lift and lower their bodies. I’ve had them recover by securing them into the proper position for a day or so, but they still waddle a bit for life because they don’t use their hocks properly.
The one small Italian chick looks like it has curled toes from the photo, this will make it impossible for it to stretch to reach the plate properly. I’m not sure how old they are, but after 2 days or so of being curled like that, I doubt you can repair them. As that chick ages, it will not be able to get around as well, and it will probably get sores on its legs and hocks from hobbling around if it lives to be an adult. I’m guessing it took a while to hatch and was one of the later ones to come out, the curled toes are pretty common for chicks who struggle inside the shell for a long time, the ankles are curled in and they’re basically in the same position they were inside the egg, and now that it’s out and growing, the rapid growth is perpetuating the issue. The chick will use a lot more energy to get around and complete basic tasks such as getting food and water, and as it weakens, will venture out less and less. If you had a bulb on them, I would recommend putting the food right under the bulb so that tiny weak chick can lay in the food and eat as much as it wants without moving or getting chilled, and I would put the water very close by.