Can we unscrew plug to run plumbing snake, or will lower part drop into drain?

Here is what you need to do... see the screws and chrome plate in the picture? You want to remove these screws and the faceplate.. and try a few things... Of course, you may find that the overflow assembly is the problem itself... can catch lots of hair on this mechanism (sorry mechanism not in picture.. but it is under the chrome plate with screws.. you will see).

1) Stick wet rag inside the drain at the upper TAPERED RUBBER GASKET area of drain... push down lightly with a screwdriver to seal the pipe. Now, fill tub with 2-3 inches of water (remove the TUB BASKET strainer if present)... then using plunger, plunge away... pull up is more important than push down.. OK? This is a great trick that only works because you are covering the overflow, so that when plunge it does not just go up the overflow pipe, but forces air/water down drain and really can clear a drain line.

2) If that did not work... use 1/4" snake or similar and snake down the through the overflow tube (tapered rubber gasket area again)... will get tough as you get into the trap, just keep pushing and will get through it... pull back and forth... in/out many times to be sure you get all the junk.

That should work for you.

Can we unscrew plug to run plumbing snake, or will lower part drop into drain?
THANK YOU SO MUCH!