I'm definitely a cat person. I used to rather like dogs, but the awful dog owners in my new town have made me turn from being okay with them to downright disliking them.
That one is either dead or in need of assistance. As you already mentioned, though, assisting comes with its own risks. Usually if a chick has problems hatching it's for reasons.
I would start with the articles here:
https://www.backyardchickens.com/articles/category/quail.32/
Feel free to ask questions after you've done a bit of browsing. :)
Coturnix quail are. There are species such as button quail that are not, though.
As long as they're dry, out of the wind and off of wire, coturnix quail can handle below freezing temperatures quite well. I didn't lose any when it got down to -7F here.
I'll admit, I hate it when I go through the 'fridge and cupboards looking for food and can't find anything but ingredients. I don't have to worry about that for a while, though, since I made 5 bread pan lasagnas last night. Had one last night, one in the 'fridge for later this week, and three...
I drive in it when it's shallow now. My car isn't built for the deep stuff. I just drive like I'm on my way to church with Grandma in the passenger seat wearing a brand new white dress with a pot of chili in her lap.
I've gotten stuck in the snow once, but I did the gas, roll back, gas, roll...
Are you feeding them enough calcium? I've found that buttons seem to be particularly sensitive if they're not getting enough calcium, and the girls die fast.
Do you have a thermometer under that heat plate so you know what the temperature is on the ground? It may not be warm enough since my understanding is that the heat plate is supposed to be very close to their backs, almost touching.
If the quail are on there, they will destroy anything you plant. I let mine destroy the plants after I harvest, except for tomatoes. The leaves and stalks of the tomato plants are toxic.
My garden beds are filled with cheap bulk garden soil (bought by the yard) and enriched with compost made...
Sometimes it takes a little while for the umbilical to break and they drag the egg around. As long as you made sure to break it without pulling on the chick, there's no problem.
I would start with the quail articles here:
https://www.backyardchickens.com/articles/category/quail.32/
Coturnix quail are ground birds, so they don't roost. They are unlikely to use a coop and prefer small individual shelters.
I keep my birds on raised beds and I move them a few times a year...