The beer is either fermenting or it's under pressure from the CO2...nothing can get in there. That's what the airlock is for. It's fine.
I'm no scientist, but I think you'd probably need many MANY batches all at peak fermentation (in the closet) to have any adverse heath risks from the CO2. Even then you'd probably need to go inside and close the door behind you, but I still don't think there would be enough produced to knock you out.
The sulfur smell will fade with time. You should be good to go.