That's a lot of questions. I'll do my best.
Leave chicks alone until mom gets off nest at least. Unless a chick falls out, then tuck it under her wing. She may be protective and not want you to touch them, she may not. Depends on hen. It's unlikely she will reject them. Never had that happen to me.
Yes, it's best she raise them, not take them from her and put in brooder, especially if she hatched them!
Ideally, she should have her a safe space to raise her chicks and not have to protect them from the other hens who may try to peck at them. So an adjoining pen or wired off area in existing pen would be nice. A straw filled dog kennel might work for a shelter to roost at night. She will be on ground nesting with them every night with chicks under her. So, nest she is in now should be ground level.
Don't take hen off nest while incubating eggs!! Once they get serious about incubating, they will "lock down" and stay on nest for first three or more days. This is very important to successful incubation of eggs, just as the last three days of incubation are very important to hatching and hen is on lockdown then too, just like we lockdown when using incubators. Heat and humidity needs to stay constant. Once their eggs get going during initial incubation, they will get off nest themselves for a short period of time to poop, eat and drink. The hen knows how long she can stay off. If weather is cool she may not get off nest, if hot, she may stay off longer.
Yes, it is supposed to be hot and damp under hen.
20 eggs is a lot of eggs for any hen to hatch imo. In fact, I think I see a brown one in front of her. If she can't "cover" them all, they can't incubate properly. I would take the outer eggs away from her, they are unlikely to hatch. How many a hen can incubate really depends on the size of the hen, but I would say for a large hen 12 to 15 is a good number. She will do all the "turning" of the eggs herself and at some point will know which ones are failing to develop and she will roll them away from the nest. It is best that other hens don't keep adding to the pile, which can be difficult to do in a hen house. So, I suggest marking the eggs she is currently on with a pencil so at least you know which ones to leave under her when you collect eating eggs. You can wait til she get's off the nest to do it or wait a few days and then reach under her and remove them one by one, mark them and set them back under her. Yes, she will raise chicks from different eggs, she really won't know the difference. The important thing is, that the chicks developing inside are all within 3 days of age development. So, you don't want to add new eggs to her "clutch" after the first three days, because "lockdown" is three days long and mom will usually get off the nest three days after the first chick hatches, so chicks younger than the others will not have opportunity to hatch.
IF YOU DECIDE TO MOVE HER AND HER EGGS TO A DIFFERENT NEST AREA BEFORE HATCH or even after the chicks are hatched, do it at night. Wait til she is truly in the broody mood, has been sitting at least a week, before you do it. Collect all the eggs from under her, put in carton, move to new nest area (make sure it's comfy) and immediately go get her and set her gently on eggs. She is very unlikely to move from that spot if it's pitch black. It's risky, because she could abandon the nest, so you always want to do it at night and make sure she is really in her incubating mood. If you can move the nest box she is already in, do that instead, it is much better to do that way, but still at night! If you wait to do it, after chicks hatch, it is harder to do that at night, so dusk might be a better time. Mom can still see and call all chicks under her before dark. If you are using a dog carrier for her new nest area, or the nest box has a door you can latch, so much the better. She can't go anywhere and neither can the chicks, so at least they will stay in their nest area for the first night. Just open the door early in morning and she should bring them back there at night again. Don't put water in there though, it may end up spilling and chilling the chicks.
Other hens do sometimes try to steal another hens eggs. Yes. Whether it's because she is going broody, or just wants an egg under her when she gets ready to lay, who knows? If Tilly gets off nest to eat or drink, that's an opportunity for another hen to roll an egg under her or get on Tilly's nest while she lays her egg. The potential problem I see with this is the chance of eggs getting broken (because of shear number of eggs) and you have a yolk mess all over those developing eggs, or hen could take an incubating egg under her nest, lay her egg, leave and Tilly's egg gets cold, because it's unlikely Tilly will retrieve it. So, I just think it's better that a broody hen has her own private broody area.
Hope that answers most of your questions. Make sure you have chick starter and a waterer the chicks can't drown in (no bowls!) and yes, it's fine for mom to eat chick starter. Mom will keep them warm at night or whenever they get chilled, make sure they have shelter out of rain and shade from direct sun and good fencing to keep predators out. Chick grit available in their pen once they get a couple of weeks old. When mom is done raising them you will be able to tell, she will get mean with them to keep them away from her (usually around six weeks old). That's when it's time to let mom go back to flock. You can either keep chicks in separate pen and let them get a little older, or try to introduce them to rest of flock, but be aware, they will need a place to run and hide because they will get pecked and put in their place as far as pecking order. Plenty of pen room is important, along with extra feeders and waterers.
