Hey.. I'm new to this mod, though I've watched it for awhile and recently set one up myself.. I'm no expert by any means to the mod, and really not to FML either - I do have significant experience in bukkit and it's mods however, and troubleshooting the game in general from a server and client standpoint.. (phew, now that that's out)...
Your pastebin contains this line:
[WARNING] [ForgeModLoader] The coremod codechicken.core.launch.CodeChickenCorePlugin does not have a MCVersion annotation, it may cause issues with this version of Minecraft
In my experience, it is ALWAYS best to get the most complex mod (GC) up and running first. THEN tackle other mods that enhance your gameplay experience. I would strongly suggest you take it back to a fresh start. Your log shows you have a TON of extra stuff/mods.. Not a great way to start a new server, in terms of troubleshooting and diagnosis.
Get your minecraft_server.jar (1.6.2). Open it up with winrar or whatever ya use, delete the meta folder, copy/paste the universal FML latest into it and run it to create all the necessary files/folders. Shut it down. Add the two GC mods to the mods folder, and restart the server again. Don't deviate from this, do not add any other stuff (yet), until you get GC up and running. THEN.. Slowly, one by one start adding whatever makes your server unique and what ya want..
I'm not saying the codechicken thing is causing problems, but I'm also not saying it isn't. It's just about impossible for other posters here to offer help when we aren't all working from the same conformity. And let's be honest - It's easier to see whether you've got the GC mod set up without other mods, then add them and see what happens than to throw everything into the pot and get frustrated cuz 'gc' isn't working. Give that a go, and let us know what happens.