First I thought I'd clear up, the chocobo racing issue was not fixed with the official 1.02 patch, only with jedwin's unofficial chocobo fix ("the chocobo patch"). As has been said though, trainers made for 1.00 won't work for 1.02 (due to the different memory layouts of the two version), and trainers made on the Windows 9x platform will not necessarily work on Windows XP due to the stricter memory access policy.
However, the chocobo patch works with either FF7 version and does not reduce "trainability", i.e. a trainer that works for the original 1.00 will still work for 1.00+chocobo patch. FF7 is available in so many versions (and languages) though, so if you really want to be able to run trainers you'd be best off trying to find an original 1.00 executable. And if that won't run on your PC, well then you're in a spot of bad luck. Have I mentioned the possibility of emulating FF7PSX instead?