Shower – bathtub drain stuck

It looks like you might be able to do this pretty easily.

To remove the stopper, simply unscrew the top of the handle of the drain stopper, then unscrew the underlying screw in the shaft to pull out the drain cover completely. Then, you'll have the cradle-thing that holds the stopper. You might be able to leave that in place (I grew up in a house that had the stoppers taken out but the "cradles" left in place, and two women with long hair would share the bath. We took turns cleaning out the drain) and place the mesh catcher over it. See this video for a demo (It should be region-free.)

You'll probably want to put down putty or some sort of long screw from the drain to the "cradle" in order to keep it in place.

First, make sure that the clog is accessible via the overflow. Try this test... While the tub is draining slowly, pour water down the overflow through a funnel. If that drains fast, then the clog is in the horizontal arm and no clearing through the overflow will help you.

What works for me is to use a wet cloth to seal off the overflow (replace it first to make as small an opening as possible). With the tub 1/2 full of water and holding the wet cloth over the overflow vent, a plunger should then be able to clear the clog because it will force water down the drain (horizontal or vertical) rather than pushing water up the overflow pipe (or pulling air down it). Should that fail, use draino or physically pry off the grate for an auger.

The plunger should pull some amount of gunk back up into the tub while you're using it, even through the grate. If that doesn't happen and the drainage does not improve, then you may have a clogged vent. You need air behind water for it to drain quickly otherwise it's like water draining from an upside-down bottle. Cleaning the vent means going up on the roof and clearing the clog from there -- a hose at full blast will usually do this.

As for fully removing the overflow and parts... I don't know.