when we cleaned the carpets here before moving in, we used Laundry soap and then after scrubbing the carpets ran it twice with just plain water to make sure all the soap was gone from the carpet, all our birds play on the floor, eat food they've thrown out of the dishes onto the floor of the floor and they're fine.
as for the waxing i'm not sure. i haven't used mop and glo, I use mr. clean and i only have one room that isn't carpet which is the kitchen and I do it when all the birds are down the hall in the bird room and when i can have the doors and windows open.
if any bird is in the living room i just use dish soap and water