: Reason I ask is that I don't believe wall mount and wood frame walls are an ideal combination (which is,
: I guess, why our house "came" with wood supports pieces shoved under a couple toilets. Also, wall mounts
: are freaking expensive. I've seen floor mount with rear discharge in Europe but never here in the US.

Wall mount toilets are more sanitary because you can mop the floor. There are floor mount rear outlet toilets but they cost more that the regular wall mounts and usually cannot be retro fit because the wall outlet must be set an exact distance above the floor.
The wood shims indicate that the nuts and washers in the wall have become set back too far or may there may not be any. In thirty years you could expect some settling requiring adjusting the carrier's face plate nuts and washers. I have installed plywood instead of sheetrock where the toilet meets the wall but you can have a wall appearance problem.
I would research removing the paint from the glazed procelain. Ask a painter's web site. LonnythePlumber