Yikes on the meds with no real reason. I've only had one experience with antibacterial/anti inflammatory, for a linnie who really was sick, and they actually made him worse, more lethargic than without. It was really hard on him. Just my one experience though, I'm sure others have had better!
For the eggs, no males needed to produce eggs, although obviously they will not be fertile if she's alone or with another female.
I don't have much experience, what has worked best for me so far is to let the egg stay on the grate overnight where she protects it (she usually lays it there in the evening or overnight), then I remove it in the morning without ado when she moves off to eat. This goes on for 4-6 eggs, one about every 48 hours, then she's done and life goes on. She does have linnie buddies in with her, but I don't have any nests or nesting material in there.
If she starts acting territorial, that's a pretty good egg indicator as well. But don't fret, the behavior will pass!
Right now mine have just finished molting and are slowly beginning to come into hormone time again.