Does this tape leave any glue residue after being removed, perhaps after it has gotten wet from showering?

Very minimal glue/adhesive, if any. we usually take it off before showering but it's gotten wet from sweat, rain, and occasionally in the shower and still peels off nicely with minimal to no residue. Sharp scissors work well when cutting it so the edges don't curl and tape on itself. Has measurements in the back that make it easy to determine size each time. Very good product vs using bandaids.

Is this waterproof as I need something that stays on in the shower?

I would not say it is waterproof, but the adhesive is very good and I take showers with it on. I do change it after showering since the media, gauze, etc. It is holding gets wet.

Picture shows a square, but description says roll - which is it?

It comes in a roll, 2" width by 11 yrds length. You can cut it as you wish: square, triangle, round, oval, rectangle, etc. Kindly, ADVANCED MEDICAL

How does this tape compare to 3M Medipore tape?

Medipore tape was our first choice but the suppliers no longer carried it. Mefix Self-Adhesive Fabric Tape has worked very well. Most other tape cause blisters.