1. There are two kinds of clay-type cat litters, clumping and non-clumping.
The clumping type forms a lump when exposed to water, and will form that lump in plumbing pipes if flushed down a toilet, so if you do this, make sure you put it down YOUR own toilet, and no one else's.
The non-clumping type will absorb some moisture, but won't turn into slippery clay. This the kind you use for winter snow/ice traction.
2. Disposable diapers go into the landfill. For those who disagree, WHERE ELSE would you put them? They DO NOT go into recycling bins.
3. Human poop goes into toilets, either flush or vault, but WITHOUT THE BAG. But you can bet money that most Americans won't do that, so the next best option is delivering it to the landfill. Landfills are trying to change over to methane digesting facilities/power plants, so that should become less of a liability, eventually.
Poop (human or dog or cat) DOES NOT go to gas station waste receptacles or into their dumpsters without their SPECIFIC permission. That is not a free service for you -- they pay good money for it, not for you.
If you refuse to use a toilet for disposal, find out where the most-local landfill, transfer station, or similar facility is, and deliver the waste to them. Yes, you may have to pay a fee. SO WHAT?
4. Urine is generally quite sterile unless you have a bladder infection or a disease like Leptospirosis (rare). A year's supply of urine contains about 2.5 pounds of useful nutrients, so you might as well return it to the soil. This does NOT mean that you should dump it at the base of living plants or trees -- it is much too concentrated and is likely to have negative effects on them. Dig a hole in the soil out in the open, empty your container, and refill the hole.