I've added creatures before but I seem to be having a little trouble adding Midgetalien's Burning Skeletons:

http://planetelderscrolls.gamespy.com/View.php?view=Mods.Detail&id=72392&id=6708

I've set their fight to 100. I thought I could just make a New creature and simply rename the ID and that's that. I've placed the skeleton in my mod, but when it spots me, it rushes over and does nothing but stay really close. If I move away, it follows me. When I attack it, it doesn't fight back, just stays close to me. Do I have to do anything else to add this properly?

I think it has AIFollow attached to it based on that behaviour. Check it under AI of that creature in the CS.

Nope. Just Wander. I also put a check next to Weapon & Shield. I'd like it to carry a weapon if possible.

I'm no expert on these, may be a script is adding AI, only NPC\Creatures with AIFollow dont attack back on hit.
Hit it more than 4 times, if it attacks back then its sure Follow, just suggesting......