By three weeks, they shouldn't need heat unless they're outside.
I don't use heat plates, I use reptile heat emitters in a heat lamp, but I would guess at 5 days plus?
That looks nice. Do you have a thermometer under the heat plate so you know what the temperatures are under there?
You're doing a good job of keeping them clean. That will help keep them healthy.
Start with the towels. You might try putting some Save-a-chick or Nutri-drench into their water. They are more likely to die, but that doesn't mean giving up on them yet. I have heard that some vitamin E in their water may help as well, but I have not tried that personally.
What are you feeding them? What are you using for bedding? This is beginning to sound like something is wrong with their environment.
Their bedding should not be anything smooth like newspaper or non textured paper towels. Grippy shelf liner is good.
I rarely leave my chicks in the incubator for more than a few hours. Let them dry out and start running around, then take them out.
For the splay legged chick, you can try hobbles, but cup therapy is easier for quail chicks because they're so small. Put the chick in a cup that is small enough...