I wouldn't call a 20x20x26 a fairly large cage... considering the amount of room they need to fly about in. This size cage would be "fine" for two birds but no more.
Exercise is important for egg laying hens... they may not be so interested in exercise when in breeding mode but remaining healthy helps keep their body in shape for laying eggs.
You may have better luck getting the parents to eat a variety of foods once the chicks start to leave the nest... chicks tend to be very curious and beaky little things... so offering a variety of foods as soon as they start to leave may encourage the parents to try out those new foods... also, parents may be willing to eat more foods while in breeding mode, so keep offering them each day! Try feeding sprouted seeds, as well as corn bread with some fresh/frozen fruits/veggies mixed in!
If you plan on keeping the offspring then I suggest you get a flight cage or two... A&E, HQ, and Centurion Aviaries all have some good flight cages that are similar and usually range in price from $80-$300 depending on where you get them from (brand new or used).
In the case of egg-binding, many suggest going directly to an avian vet if you believe something is amiss.... although some have suggested holding the hen over/in warm/steamy water and gently massage their rump (being careful not to break the egg) and others have suggested giving oil (goes RIGHT through their system! I know first hand, as I had a young [at the time] female tiel drink a couple gulps of cooking oil... obviously, she had TOO much and was not well for a couple of weeks after that)
As for handfeeding, I suggest you try getting some experience from a reputable breeder, handfeeder, or avian vet.
As for monitoring the nestbox, well you don't want to disturb them too frequently, so it would be best either when the hen is not inside the box (assuming the box is on the outside of the cage where you can easily reach it) or checking when they are least active... you don't need to check more than once a day (or twice a day), but less frequently (such as every other day) would do just as well.
Handling the chicks varies... some say you can handle them the day they hatch while others say that you shouldn't handle them until they are two weeks of age or older... the reasoning behind this is that younger chicks get colder faster... therefore it's recommended to keep the room the birds are in extra warm. If you handle them early you probably don't want to handle them for any longer than 5-10 minutes... but when they start feathering out you could probably handle them for 20-30 minutes at a time.
The crops are easy to see when they have no feathers. Just born chicks will be able to survive for some hours before feeding as they still have food from when they were inside the shell. I don't know the time frame from when chicks hatch to when their parents feed them (this may vary) but they should be fed within the first 24 hours after hatching.
I'm not experienced in breeding in any way... I've tried breeding budgies with no luck (two pairs, either no eggs or infertile eggs) and bourkes (first clutch infertile, second clutch two fertile eggs). I've never handfed, but understand the gist of things (and this is not including feeding a 13 year old mitred conure handfeeding formula!

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