1- Taba’iyyat means when the najes water is drained, whatever is inside a container becomes tahir. That’s how cups are purified with Qalil water. And yes the lid becomes pure according to this principle.
2- However, if there are clothes inside the washing machine and you know pure water didn’t reach them at all, then they don’t become pure. Water must reach them.
3- As for the detergent froth, since pure water ran through it, it became pure.