My plumber had to knock out some of my bathroom wall tile to access the stack and make a repair. I am trying to go about repairing the hole. However I cannot separate the tile from the existing wall board. I believe they used some sort of adhesive wall board because I am not encountering thinset. Should I just cut all the way through along the grout lines and butt the new cement board up to the existing wall. I would be unable to tape those joints then. What is the best way to go about the repair? Attached is a picture of the main area of the repair. Note this is on the opposite wall of the shower and does not get direct water exposure.
Your plan is sound. Cut the damaged tile out along grout lines, being very careful to not chip the tiles that will remain.
Install new framing backing where possible, using construction adhesive to bond the original substrate to the lumber. Then, when you install your new backer, everything will be tied together well and taping the joint won't be necessary.
Clearly this is an old house and you are somewhat lucky. On a new shower you'd expect to see the shower liner / waterproofing coming up the wall. It just looks like tile over drywall which is pretty terrible in the long term for mold growth and the wall rotting away (especially walls that get wet with showering).
If you like the shower / tile and can get replacement tiles that match existing I'd just add nailing edges, concrete board, a liquid water proofing membrane something like:
then add your tile with thinset and grout it.
I'd either use a oscillating tool to remove the grout, then cut away the section or go with a diamond blade in a circ saw and setup a guide to get a nice cut without chipping the tiles near your patch. Wear a respirator. Drywall mud contained asbestos at certain points in time and the dust from concrete / tiles / grout / drywall won't be good for you in any case.
Behind the drawer I'd just get a 1/4" piece of mdf and put it at the back.
Also this looks to be an exterior wall given the insulation so it wouldn't hurt to fill in areas that aren't properly filled with insulation. I'd swap out the fiberglass for roxul at least in the shower area(as it continues to be effective even if wet).
Depending on your climate zone you need to think about vapor barrier - though it doesn't look like there is an existing one and throwing a small piece in probably doesn't make a lot of sense. The original paint in the house was likely serving as the vapor barrier.