Breezy-As for your cat's breath, maybe it has a cracked tooth or something stuck between two of them. My dog had the stinkiest breath until I took her in and had her teeth cleaned. Then I stopped feeding her the canned food daily so that the tartar wouldn't build up again.
Your cat may be pooing on the rug in front of the box because it hadn't been fixed. That problem may disappear after having kitty altered. I keep a long flat tray in front of my boxes, (like the kind that come in the bottom of dog cages). It keeps the litter from getting all over the place and I can just pick it up and dump it into their boxes when I clean them.
I changed the type of litter I use and it seemed to cut down a lot on the smell. I buy Fresh Step Scoopable now. It makes it a lot easier to clean and not so stinky. I don't think my kitty knows about covering her litter, but the older cat does. I think either she is teaching the kitten or doing it for her. LOL! Who says you can't teach a stray cat new tricks!