Touching the soap doesn’t matter at all. You’re about to wash it all off your hands.

Soap and water definitely work to reduce infections, even when using a shared sink. Look up what happened in the 19th century when people finally figured this out and started hand washing regularly, infectious disease transmission in settings like hospitals dropped dramatically.

In the 19th century infections were cut drastically when hands were washed with water only. Imagine what soap can do.

They started from not washing hands between autopsies and surgeries though. Talk about low hanging fruit.

I believe I did answer your question. Washing your hands will always be better than not. Many modern buildings also have automated water/soap/towel dispensers. And you can always use the towel to open the door.

Yes, but on your way into the stall, you touched the stall door, and then you touched other things, including your clothing, which you didn't wash on the way out, only your hands.

Germs probably survive better on your skin than on the cold metal tap in the bathroom. Food safe handwashing says use a paper towel to turn off the tap after you are done drying your hands with it.