We raise our quail on the ground in a moveable coop/run set up. Having access to fresh grass, plants, bugs, etc is how mother nature intended (not confined to a small cage in my humble opinion). In the Fall they really loved being moved into our garden (lots of plants they hadn't experienced before). As far as I know, there shouldn't be an issue just in your yard. Check the chicken section for poisonous plants, wouldn't hurt to follow the same rules. Also, weed killer is a no-no.
As far as parasites & viruses, you are always going to run that risk unless you keep them in a completely sterile environment. You can't stop wild birds from pooping in your yard & you can't stop other animals from pooping there either. We're on a farm in the middle of no where, it's a risk that we've accepted. Since they have short lives & we haven't noticed sickly behavior, we don't bother doing anti parasite medication.
The heads in the air, mouth open thing...do they look like they're breathing fast? Like hyperventilating with a swallowing motion? That sounds like stress, maybe over heating, but most likely stress. We've seen ours do that when they are super freaked out (first day in the coop, they missed their brooder for a few hours) & more of a panting action when it got over 100 degrees in the shade (frozen veggies, frozen water bottles both helped them get through the heat).
For eggs....Do you have males & females? Make sure you have the proper ratios to avoid fighting & too much love making. Assuming you're good on that, yes they can fertilize their eggs. What breed do you have? We have Jumbo/Coturnix & ours at least have no interest in going broody/hatching eggs or looking after wee ones. I'd suggest you collect the eggs & incubate them yourself if you're looking to maintain your flock that way.
