yes, they can double clutch, though it's not recommended to take the eggs out right away, because of calcium deficiency. If one has to remove the clutch of fertile eggs, best way to do it is replace them with fake eggs.
A normal diet for the parents is ok when they're raising chicks. Just have to make sure they have all the nutrients they need, since some of that food is going to the baby, as well.

If for any reason though, you have to feed the babies yourself (if the parents got sick or won't care for them), you can find some chick-food at pet stores. You would need a certain kind of syringe with a very narrow tip, since zebra finch babies are SUPER tiny. I'm sure a dropper would work well, if it was small enough.

I would just leave the work to the parents, though. Hand feeding isn't really recommended unless if it's absolutely necessary for the chicks to survive.