Hi Taeung, and welcome to SoCalHiker!
1) Yes, I carry out dirty toilet paper. I use a sandwich sized zip-loc bag to hold the dirty TP, then put that bag inside another gallon-size zip-loc bag (just in case). The exception would be at pit toilets. Toilet paper can go in (but no other trash).
2) Yes, but usually only at night. During the day, my “trash” bag is in an outside pocket of my pack.
3) I don’t carry soap, but I do carry body wipes and hand sanitizer. You can wash the sweat and dust off in streams and lakes without soap, and then augment that with the body wipe and/or sanitizer as required before handling food.
Note that within the Mount Whitney Zone, you not only have to pack out your toilet paper, but they give you special “wag bags” to poop in… and yes, you’ve got to carry that out. So far, I’ve avoided having to use them.