If just a few of the poops are more brown and liquid, the probiotics should take care of that.
Medicated feed should be taking care of most cocci issues in such a clean environment, so I'd temporarily check that off the list as well.
The sheet - I use papertowels, so similar. I just wondered if there was a scent/chemical/fragrance that would be causing the sneezing. And yes - I always start my babies on something like cloth, or rubbery shelving liners, because they are less apt to splay their legs. And because of the eating issue. My babies usually start with a shallow wide feeder as their first til they figure out that whole eating thing.
Then I go to wire or shavings after about a week. So that's all fine.
About the other man, or anyone really supplying eggs, birds can come down with some respiratory illnesses and survive. In surviving, they become carriers. That's not necessarily as bad as it sounds if the flock is a pet flock and you keep your eggs for your own birds and understand that if could crop up again. However, anyone selling hatching eggs and breeders has to understand that they're selling the disease if it's one of the ones that passes in the eggs (like M. gallisepticum does). So even if a flock doesn't have respiratory problems, that doesn't mean they never had.
Many fanciers who are serious about their chickens will get a sick bird tested from the flock and then decide whether or not to cull the chicks based on that result. Testing is a good idea for the more casual fancier as well (so you know what you're dealing with) although culling isn't as much of a necessity if we again keep a closed flock, don't breed for sale, and don't sell eggs for hatching. And then again - it could come back to haunt you, in theory.
As for them being the first flock, that's good - you don't have to worry about taking it out to another bird. Note that you can also bring these things home from visiting a neighbor's flock. And wild birds can carry them as well. But since these re inside... o problem there.
If they were mine, I'd watch them carefully today. Plan a trip to the feedstore and
Walmart tomorrow to pick up the goodies. At the feedstore, look for a small bottle of VetRx. It's not amedication but a breathing aid and I feel it has a small antibacterial action topically. It's great as a tonic for birds with respiratory illness and helps keep their sinuses clear so they don't get infections.
It's possible that these babies had a little bit of a disadvantage, immune system wise, from the bobble in the hatching. But it's a good sign that they all did hatch - strong babies. So who knows..
Watch the poop - you might have Corid and/or Sulmet on hand just in case. I would also consider a Duramycin package to have on hand, although normally I'm certainly no fan of the stuff or antibiotic treatment until you know for sure that what you're seeing is bacterial rather than viral. Antibiotics and young chicks, together, are problematic.