I have some files with the extension .S3M. They're kind of neat. Are these a variation on the mod format?

Yes. S3M files are produced with legendary Scream Tracker. That's a PC/MSDOS module tracker, something like 7-8 years ago. All those mod formats refer to tracker based music, which means that you have a set of samples that you can use and play them on different frequency and apply some effects to them, such as panning. Basically, they are like midi music but with a custom instrument set in each file (and with a lot more twisted effects than in midi files).

Another question about mods: Are they supposed to sound identical regardless of the player, or will they sound different if the player implements the playback differently?

Basically they should sound identical regardless of the player, but that's only in case that players would implement all commands/features correctly. Different kind of pannings, etc. might be implemented differently and some players might lack some commands and so on.

Oh, those days were the golden days :)

Kim