Hi Scout, and welcome to the wonder world of finches! You've chosen two of the "easiest" breeding finches, which means you are in for a great deal of work keeping the population under control -- but a great deal of fun watching your guys & their offspring!
Leaving all your guys in the same cage depends on a couple of things. First, how big is the cage? Finches do need enough room to move around & feel they have some territory of their own, so unless it's a very big cage, you will probably want to do some separating.
Second consideration is, of course, inbreeding. Left to their own devices, those sweet little baby birds will soon start hooking up and making MORE sweet little baby birds. Zebras, in particular, are notorious breeders so unless you want to have a whole flock rather quickly--and an inbred one at that--, you'll want to put some time & effort into separating your chicks.
I know catching the little guys is tough but if you're patient, you can manage it. When using a net, take all the perches, nests, etc. out of the cage -- gives them fewer places to "hide". Often, if you just put your hand in the cage and hold it very still, you can eventually grab somebody as he flits by. If none of that works, wait til night. As you already know, finches are very still at night; in a darkened room, using a flashlight as necessary, you can usually just reach into the cage and pick them up off the perches.
As for birdie birth control, we always made a point of housing our baby finches in separate "dormitories" for hens and cocks. The parent pairs remain together, of course, but we do constant "egg patrols," removing new eggs every day or so. We sometimes save the old eggs til we're sure they're no longer viable, then substitute them back for the fertile new ones we're stealing. That gives mom & dad the opportunity to tend a clutch for a few days, which slows down the laying process.
If your hens are laying regularly, make sure they get plenty of calcium in their diets or you may have problems with egg binding -- the eggshell doesn't harden and the hen can't pass it. Just yesterday we had to "hospitalize" one of our Gouldian hens who was egg bound. A small separate cage, some warmth (we use a heating pad on low - medium heat) and a little mineral oil rubbed on the vent work well for us; Pepper will be rejoining her flock later this morning.
Hope this helps. Let us know how your babies mature!
Cheers,
Pat