C&C3 version 1.09 doesn't like me. Please Help!

That is the game executable, yes. But it is normally ran through a launcher, since you can't launch executables directly unless their extension is .exe; renaming them to .dat is a classic way games use to force users to use the launcher instead, which often gives extra configuration or options to the game. (It's not new. The original 1995 DOS C&C1 has "game.dat" as executable, ran by a tiny "C&C.exe" launcher program)

For example, in C&C3's case, the game's launcher can select a version from the different cnc3game.dat files in the version folders.

But yeah, it's very possible that feature was broken by Origin, then :-\

I'll look further into this, and see if maybe it can be repaired by adding the older version folders manually or something. But if these .dat files themselves were adapted to Origin, that chance is rather low.