If I couldn't mop the floors we'd be in serious trouble here! I buy a special cleaner that's hypoallergenic because of my husband's chemical allergies. It works great around the birds (although I have not tried it on their cages). It's called SafeChoice Concentrated Super Clean (All purpose cleaner and degreaser). No fumes, no perfumes and it really works. You dilute it according to the type of use you're putting it to (spraying counters, mopping floors, etc).
www.afmsafecoat.comI try to avoid anything with perfumes or strong chemicals. I do use essential oils from time to time and haven't had any difficulties so far. I threw out about all of my teflon coated pans, but I hung onto the bread machine and rice cooker (that have coatings on the inner parts). I simply plug them in outside the backdoor so they're not in the house. If you don't have that option you might open all the windows, turn on a bunch of fans. I prefer to play it safe and just not take a chance because there's safer alternatives.