Yes there are subtle differences between the different #?Tracker formats but they will still play. ProTracker added a number of extra commands that SoundTracker didn't have so certainly a lot of ProTracker mods would sound wrong in SoundTracker, but you could still load/play/edit them there. It was essentially the same format with minor differences in implementation. There were expections to this, SoundTracker 2.6 for some reason didn't use patterns, but rather had a different sequence for each channel. That was very odd and rather inconvenient.
Of course, you can use (Octa)Med music in games. You can use anything, there are games that use music written in Aegis Sonix but I wouldn't recommend it if performance was a factor (or if you wanted your music to sound any good). SoundTracker became something of a standard in the demo scene and games industry very early on because of a lack of viable competition. Faster formats have appeared since but aren't as well supported. (Octa)Med wasn't designed to be fast, it was designed to sound good and have lots of features.