I've just stumbled across something:
There are two files called 00_CCTDRDXINI.tdr and 00_CCTDRDXINI.big. Both files have the same size. It is supposed to be that way?
Edit: Luke1 and I have done as SgtMyers88 instructed us to do: Delete all TDR-files and update Tiberian Dawn Redux.
And we've just played another game of TDR via GameRanger and we haven't had any of the problems I reported earlier.
The game has run fine for about five minutes, with some minor connection issues. After those five minutes, a mismatch has occured.
However, my guess is that this has been Zero Hour's typical mismatch issue and that it hasn't been caused by TDR.