Does something terrible happen if I apply the patches out of order, or does the patcher check to make sure that the prerequisites are in place before running?

The patches check and give an error if they can not update the installed version (IIRC they just say game not found, or something, rather than list the found and expected versions or any prerequisites), the patch has already been installed (usually, a couple will install again over the same version), or the game version is newer than what the patch would apply.