If your kitty is new to your home, you may be feeding him different food than he was eating before. Cats have very sensitive stomachs, and switching foods should always be done slowly or they'll likely get diarrhea. Sometimes the stress of being in a new home results in diarrhea. The kitty needs to see a vet to make sure he doesn't have worms or other parasites, but stress-related diarrhea is quite normal.
He's wiping his butt on your floor after he goes because that's how he cleans himself. It's not like he's got access to toilet paper and hands! When our kitties have diarrhea, we make sure we keep an eye on them and when they go to the bathroom, we pick them up after they've left the litterbox and use a warm wet paper towel on their rear ends before they get the chance to wipe them on the floor.
When you clean up the poop, you must use an enzyme cleaner to ensure that you remove ALL the smell, or your kitty may get confused between the litterbox and the carpet where he wiped. You can pick up an Enzyme Cleaner such as Nature's Miracle or Simple Solutions at any pet store. Simply follow the directions - it's easy. With warm water, clean up what you can. Then shake the bottle really well, soak the carpet down to the floorboards, let it sit for 10 minutes, then blot it up and let it dry. We've never had a problem with stains, and we've never had a problem with the cats using the same place to go to the bathroom again.