How far is it from the nest box to the bottom of the cage?
If it is very far, then put a soft towel at the bottom of the cage below the next box opening in case the baby falls.
The Dad will continue to feed the baby until it is fully weaned.
You need to have a millet spray near the nest box as well as seed, vegetables and water.
The baby will start trying to crack seeds on his own and when he stops "begging" to be fed by the Dad, then he is fully weaned.
Yes, it is normal for the older babies to feed the younger ones
Since the Mom and Dad are starting to try to mate again, you need to separate the MOM into a different cage on her own and put it in a different room from the Dad and the babies.
As she is becoming broody again already, she may well become aggressive toward the chicks and you certainly don't want her to double-clutch.
Leave the Dad in with the babies as he is the one that cares for them until they are fully weaned.