TP and a few paper towels/napkins are standard in my cars and portable kit. When you need them, boy do you need them. I also carry a poly tarp so I can create a bit of a shelter, workspace, or private potty corner out of the wind.

I carry hand sanitizer as well, which is multipurpose: hand cleanup, personal tidy up, infection control, firestarter, gas stink remover (use lots, then scrub hands with snow, repeat).

In rural areas, I wouldn't worry about the waste unless it would wash directly into a body of water. On a heavily travelled route, I would let solid waste freeze and scoop it into a doubled trash bag. Not really different from cleaning up after your dog.