If I don’t want to use this for it’s intended use (near toilet), can I remove the shelf-like thing that holds the plunger so the space is all open?

Only on the hydit200.

What are the rough dimensions?

The rough opening is 9 1/4″ x 25 1/4″.

Is the door swing reversible? In other words, can the door be switched to swing either way?

Yes, the door can hinge left or right.

Could the door be painted to match my wall color so it blends in better?

Yes it can! Feel free to paint it the color that most suits your bathroom.

The plunger in the pictures is a sink plunger. Will a real toilet plunger, with the bottom flange, fit?

The plunger provided is a 6″ toilet plunger, not a 4″sink plunger. Plungers with the added bulb on the bottom do not fit very well. Been using the provided plunger with no issues for years.

Where is it made?

The current hydit100 was manufactured in China. The new hydit200’s will be manufactured in the U.S. and will be available in the next few weeks.

How much space is needed for the door to fully open?

The width of the door at a minimum. Door is approx. 12″.

Does installation require cutting into the wall?

Yes, rough opening is 9 1/4″ x 25 1/4″.

What is holding the plunger- is the dirty yuck water going to just drip all over the place and cause a very hard to clean situation?

The plunger is held by compression in the restraint. The bottom of the Hydit100 has a depression to retain moisture until it evaporates. A wipe once in awhile will keep it clean.