Is trodden on like stepped on?

So if children always skipped with a rope, you call it a skipping rope, even if nowadays you can get them made of plastic or metal cable, and with modern materials some might be quite thin. The material has to be strong enough to survive scraping along the ground and being trodden on, as well as heavy enough to turn evenly. So it is acting like a rope. Once it gets its name, the name sticks!

Tara2Is trodden on like stepped on?

Yes. It is exactly like it. "Tread" is pretty rare in spoken US English, by the way, outside of fixed expression like "tread lightly".