They are indeed stored in files, namely those files in your <ME_PATH>\Config directory with extensions of .tpt. However, these are database files, very sensitive to changes, and containing control characters. While you can hand-edit such files, it’s not generally a good idea.
Fortunately, in this case it isn’t necessary. If you open Tools | Customize… | General | Templates… from the main menu, there are controls on the Edit Templates dialog that allow you to move templates up and down in the list for any given language.
One useful trick you can get a lot of mileage out of is to name your templates in such a way that you can select any particular one by typing a minimum of keystrokes on the drop-down list you get by pressing Alt-F9. For instance, if you have a lot of entries you’ve named with the same first ten characters, you’ll have to type at least that many to reach a unique combination. If, instead, you rearranged the name to put the unique part up front, you’ll reach it in one or two keystrokes, ideally. You can also use special symbols (so long as you avoid the ones that mean something to Multi-Edit, such as @ and #), numbers, or one- or two-letter prefixes to group items for quick key entry. The set of templates I’m using to do the formatting in this message uses that technique, actually. You just have to be careful not to let the names become too unintuitive, and to actually fill in the description box for each template.