What is the width of Giant CR41, 32-hole rims
I went and bought a ruler. It's 13mm.

I'm kind of surprised they'd stick a 35mm tyre on a 13mm rim out of the factory. I'm going to stick 700 x 28c conti GP 4 Seasons on it unless someone points out a clear reason not to. Sheldon Brown's guide says 25 should be a max for that rim, but it also grants a bit of room for error and this rim was fine with 35's...

This also means that CR41 isn't necessarily a material/feature code and it's possible the TCX and TCR use the exact same rim.