So I bought the DA:O game from GOG, and it has a total of 8 files to download. Six of them are ".bin" and two are ".exe", so my question is, in which order should I run those .exe files? Does it even matter? Yeah I'm probably just a paranoid noob, but I appreciate any feedback anyone can provide.

IIRC run 1st : setup_dragon_age_origins_ultimate_2.1.0.4 and then patch_dragon_age_origins_ultimate_2.1.1.5 (hopefully file names are the same for you)

The .bin files are part of the installer -- splitting it up into multiple files is done to prevent having to download massive files. kokosabre is correct: you need to run the .exe file for the main installer first, then the .exe for the patch. Running these in the wrong order won't work, but won't screw anything up either. You can't patch the game until it is installed.