"Normal" people don't mod Morrowind.
Anyway, using a mod manager like http://www.gamesas.com/topic/1180562-yacobys-wrye-mash-fork/ to install mods can make things easier in the long run, because you will be able to keep track of which files are from which mod and can uninstall them again if something goes wrong. Your pastebin guide doesn't seem to mention that, though.
Also, for the most part, the dates don't matter, as there can be only one file with the same name in a folder. But...
...this possibly refers to one of two things that come to my mind. First, replacers. The original Morrowind game files (meshes, textures, etc) are stored inside the Morrowind.bsa file. To replace a file inside the .bsa, a file with the same name has to be placed in a folder that corresponds to the path of this file inside the BSA, so to replace for example "textures\tx_wood.dds" inside the BSA, there has to be a "tx_wood.dds" in the "textures" folder in Data Files. But this file also needs to have a newer "date modified" than the .bsa that contains the file it is intended to replace. For non-Steam versions of the game (don't know about the gog version, though), the .bsa is dated to some time in 2002, so basically all modded files are newer. The steam version gives it the installation date, however, which means it has to be changed for mods to work. If I am not mistaken, the Morrowind Code Patch takes care of this (and the .bsas of the expansions), which you should be using if you followed the guide.
Second, Plugins. Unlike the later games, Morrowind loads master and plugin files (.esm, .esp) according to their modification dates (although .esm files are always loaded before .esp files). That means, if two plugins change the same thing in the game, the newer one will be loaded last and the change it does will appear in the game. The mlox tool is intended to sort plugins in the best possible way (by making sure that important changes from one plugin are not overwritten by other mods), and it does that by modifying their dates. Also, Wrye Bash can be used to change the load order manually, and it will set the dates accordingly.