Recalculate your normals, and re-export the model.

Don't apply the edgesplit modifier. There is no need. It's called edgesplit because it splits edges. It only does so when you 1) apply the modifier or 2) export the model. So there is no need to apply it since it automatically applies when it exports.

I really don't know what the issue is. Try studiocompiler: http://www.chaosincarnate.net/cannonfodder/cftools.php?program=studiocompiler

It has a decompiling tab. Check that website on how to use it.

Hm... The new steampipe system seems to have messed the Source SDK up really badly. Try putting the decompiler program in your
C:\Program Files (x86)\Steam\SteamApps\common\Team Fortress 2\bin folder instead. This might work.

Did you try refreshing the source SDK content? Try that, then retry the decompiler.

Careful with the color pallet. Red isn't usually associated with necrolyte. I would suggest changing the red to something else (maybe black or blue).

I don't know what is the problem... does your model work in hlmv? There are so many problems with itemtest, I wouldn't bother using it.

Load a player model into hlmv, then go to file > (steam) load weapon > find your hat > check to see if it applied properly. If it did, then you should be fine.

You are not the first person to ask me about this... apparently many people are having this problem. I would contact Valve or the TF team (their email is on their website).

The shoulder and bracer look amazing in-game. I don't think a laser sword is a good idea, though... perhaps you could just make a blue blade. The laser sword will most likely get rejected because Valve doesn't want any copyright issues with lightsabers.

No.