Hi! Sorry to be a tad late getting back to you...was just on four days off...
First, I think it's physiologically impossible for any bird to lay more than one egg per 24 hours. There's quite a journey involved for every yolk to acquire several coats of 'white', the stabilizing 'cords', membrane, shell, shell pigment, etc, when you think about it, and I suspect that it probably depends upon each female's individual condition as to how well she can generate all that requisite material day after day. Most of the little finches manage an egg a day, but I'd never be surprised if someone skipped a day. Individual condition and just the general fitness of the reproductive structures within likely also determines how wearing the egg-laying process is...some birds seem to squeeze out their eggs with hardly a blink, others look pained and worn for hours beforehand and vastly relieved afterwards. Even humans vary in this respect--we probably all know women who went through hours of agonizing labour and others who just seem to pop out their babies as easily as squirting out a watermelon seed! It's probable that your gal was still feeling a little worn by her bout with diarrhea so was more affected by laying than usual. Zebra finches, in my experience, usually lay easily, though I've had a few older matrons that would show the stress and who needed limiting to one or at most, two clutches a year.
As for handling eggs, as long as your hands weren't horribly sweaty or recently liberally smeared with cold cream or something, the eggs should be fine. You're right to worry about this sort of thing, though. It's so easy to maybe touch a chemical, perhaps use a bug spray or lotion, then forget about it and hours later unwittingly transfer a still-lethal dose to a delicate organism like a pet insect, tropical fish, or a tiny finch egg. I usually rinse my hands in hot water (no soap) beforehand just in case if I handle little eggs. Better yet, if I can safely get at them with it, I'll use a little spoon shaped like a soup ladle to move eggs, but obviously this really only works well if you can see the eggs well enough to be sure of not accidently rapping and squishing them with said spoon!