This is usually caused by starting from a save file rather than a new game; the initial state for all characters (starting weapons, materia, etc.) is done when selecting New Game, so if Red XIII doesn't have Enemy Skill it's likely due to this. You'll likely be missing Barrier, Enemy Lure, Earth, Heal, Revive, Destruct, Gravity, and an Elemental Materia in addition to Yuffie's Added Effect, Time, Exit, and Enemy Away.
The starting stats of characters will also likely be wrong, and Tifa/Barret probably started with Leather Glove + Gatling Gun instead of Metal Knuckle and Assault Gun which were repurposed into triple AP weapons for later in the game.
Best way to sort this out is with a save editor. Add any of the missing Materia to your inventory (Barrier Materia should be given 5000AP as it starts with this to have MBarrier unlocked). Also, change those triple AP weapons to metal knuckle/assault gun if you have them.
Starting stats of the characters can also be changed safely, as handily they do not change upon level up (a different system involving Sources is used to raise character stats). I've written them in below, in order of STR, VIT, MAG, SPR, DEX, LCK:
Cloud: 43, 28, 27, 32, 16, 14
Barret: 45, 43, 9, 37, 8, 12
Tifa: 47, 12, 23, 18, 42, 9
Aeris: 22, 11, 47, 22, 32, 25
Red: 35, 27, 32, 25, 35, 12
Yuffie: 37, 8, 38, 11, 50, 12
Cid: 50, 34, 12, 10, 11, 90
However, if it was just the Enemy Skill that was missing and everything else mentioned was correct then maybe it's an older version where Red didn't have enemy skill? The catalog version is 1.4 and I can't remember all the changes from then and now.
Thanks for the detailed reply. I ended up starting over since I had most likely installed NT incorrectly. I seem to have mostly everything working now, by using a combination of the stand-alone installer plus a few mods in 7H (60 fps fix, music, and textures). I only tried the NT IRO + 7H method once, but the mod wasn't installed correctly and all of the character's base stats were off. I'm guessing it's because 7H can be so fussy and something went wrong along the way.
Anyways, I think I'm having an issue with materia's stat effects: The NT features list mentions that materia stat effects have been altered. Materia benefits/penalties don't appear to be changing for me even though everything else seems normal for NT. My basic magic materia (Fire, Ice, Lightning) are all showing the standard stat penalties (-2% HP, +2% MP, etc). Just to clarify, I'm currently using the stand-alone installation (the may 18th update) and 7H with the 60 fps fix, music and texture mods installed.