The cooper's hawks here like to wait until they get pullet size with some meat on them. Encourage blue jays if you have them, as they will sometimes chase hawks off, probably mainly when they are raising young. They mimic the sound of the hawk. I have to check to see whether I'm hearing a hawk or bluejay sometimes.
As to when they can eat bugs, I have a broody with 8 three day old chicks, and I saw one find it's first tiny worm today. Made a soft version of that sound that roosters make when they've found something yummy for the girls. Mom had already been scratching up bugs for them, and showing how to dust bathe. Don't think they caught on to that yet, though she threw some dust their way.
I have an exceptionally docile flock at the moment, and am spending most of my day working outside and supervising. Also, we're having nice warm Ga. weather, and there are tons of hiding places. The other 25 hens, 2 roos, and 1 guinea have 10 acres with lots of things to occupy them, and just come in to lay an egg. They haven't shown a lot of interest in the chicks.
The mama and chicks do have a nest crate with a screen topped dog exercise pen around it within the chicken house. But I'm letting them come and go. A couple hens did wander in and lay an egg in the chicks' nest, but nobody got upset. This mama is a very easy going marans.
The chicks don't appear to get cold. Mama takes breaks to go back to the nest where they get under her, but not as often as you would think. Sometimes Rufus, the guinea, guards them, even though he's a pain in the tailfeathers to everyone else. Last year, one of my mother hens died while her young were not quite old enough to be on their own. Rufus was very attached to that hen and been hanging out with them. He finished raising them just as the mother would have.
I spent most of last summer trying to keep broodies and their chicks captive, constantly refilling and cleaning knocked over pooped in or dirt filled waterers and feeders when the mama showed how to dig and bathe, and cleaning mess on the ground, with mama hens that tried to rip my arm off. I found that when they escaped they did just as well on their own with the chicks. It seemed the chicks developed faster, too. I do have another broody setting, and she and her chicks will have to stay separated so as not to injure the others.