How to remove a plunger inside the overflow assembly? without the rod

Unfortunately, I think you're fixin' to go for a crawl with the crawlies. (At least you have that option -- the same task in my house would require a jackhammer.) There SHOULD be a P-trap below the drain. If you can get the tailpiece off of the bottom of the tub (unscrew the metal drain, then the tailpiece should pop out), you should be able to spear the plug.

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.