In a small flock, yes, but in large flocks (10+) I would never allow it.
She is brooding which means she is going to try to hatch eggs into chicks. Some of the other hens will lay eggs near her and she will roll those eggs under herself. If you have a rooster in the flock and want some chicks you should let her stay where she is but if your flock has no rooster then take the eggs away from her.
Roosters make this noise to calm and reassure the hen all is right with the flock. Roosters have many ways to communicate with the hens such as crowing, which is a call to other roosters to let them know he is guarding the flock or to let the hens who have wandered away from the flock to come back.
They need a mother until they grow feathers which might be in about a week, then you can let the mother be a normal hen again, but separate chicks from all birds
Just let her be. The newly hatched chicks will take up some of momma's time and the room under her but the mother hen will adjust things by herself. Trying to do this for her will just stress her out and may cause her to leave the nest. The new chicks will find room under her wings eventually and mother hen will roll the unhatched eggs closer together.
Yes. Momma hen will keep the chicks safe and protected for about 8 weeks after hatch as long as she is able to keep them near her. Chicks however are curious and will explore their new world. This can be hazardous and other members of the flock will peck at them.
Not as a rule. If you have seen a hen eating a chick it is likely another flock member cannibalizing a recent kill. Mother hens are usually very protective of her chicks but other older birds in the same flock can peck at a new chick thereby killing it, once dead, a chick is just another source of protein to the other flock members. The scent of fresh blood will attract the others. If a chick is killed soon after introduction to the flock you should be sure to remove its body and dispose of it. Allowing the flock to eat their own is not something you want to let them do.
There really is no need. The hen will turn them herself when she's laying on them. She won't let you touch them anyway when she's sitting on them.
Leave them alone and let the mother do her job. Your hen will continue to lay on the eggs a few days after the first ones hatch. The chicks will tuck themselves up under her to stay warm and the hen will keep them and the eggs warm while waiting. Good luck!
The only way to accomplish this is to separate them until the chicks are old enough to stay out of harms way. Allowing the chicks to socialize with the flock can be done with a temporary cage, the chicks are protected yet among the older birds. All flocks of chickens have a pecking order and there is no way to safely stop this, the more grown up your chicks are, the better able to handle the pecking and chasing until peace reigns in the hen house.
The rooster is often not the problem. The dominant hens are usually the aggressors when it comes to chicks. Is the rooster mean to the hen or chicks, if not then no, he will be just fine. If the rooster is mean then yes, but remove the brood hen and chicks to a safer location, not the rooster. A rooster is dominant when it comes to his territory but usually ignores the chicks running around.
Do nothing. Leave her be and let her do what comes naturally. The is no need for you to "help", in fact helping her at this time may be very bad. She will be protective of her peeping brood and can get quite aggressive biting/pecking at your hand. Let her alone and she will soon have a bunch of fuzzy little peepers following her around the barnyard getting into all sorts of rough and tumble trouble. If you really want to help, keep the dog and the cat in the house for a few days or at least when they are in the yard, as pets can be the new chicks worst nightmare.
Yes. Move them at night and try to minimise noise and disturbance. Put the eggs into the new nest before putting the hen in and don't let the hen see you taking the eggs out of the old nest. There is always some risk she will stop sitting depending on disturbance levels and the personality of the individual hen but more often than not you can move them successfully.