Amumu can also build Everfrost (If you haven't already checked that out, try it!) which gives him a ton of utility. Having Everfrost available more often feels really nice.
Both Ultimate and Treasure hunter subrunes work well, indeed. I really think this is a matter of preference. As you become more adept with Amumu, you should get a good idea of what subrune here works the best. I'll look into adding a little paragraph about these options in the guide!

I've not tested anything so I can't say for sure. Based of my observations,
I don't have any comments on Demonic Embrace, it's probably worth waiting to see.
Abyssal Mask will really only be a buy into AP teams. I don't think the MR debuff is going to be strong, the number is not mentioned.
FoN should become significantly stronger with priority.

Also, Cosmic Drive should feel really good for Amumu's AP build, even though he's not a fighter, the MS should help so long as it stays 30%+.

Hey there!
You don't ever start with your W because it is 1; mana consuming (which doesn't matter if you start blue) 2; it doesn't do enough damage, especially to clear the camp.
People like to start with their W & Q. Neither of them clear the camp that quickly, your E will do it the fastest. Tantrum(E) will reduce its cooldown every time the blue attacks you, making it also more available. The longer you spend at a camp, the more damage you probably will take. I'm looking at amumu's damage and it costs 8 mana per second (for base), and you can keep it active for about a little more than 36 seconds before running out of mana. It deals 10 damage per second, while his E will deal 75 damage every 10 seconds, reduced by an entire 0.5 seconds every attack you take.
To summarize, his W is mana-costly and you can use some more damage to clear the camp quicker. In terms of damage, they fall close to equal in damage (his E probably does more however), but his E uses less mana, and even though you'll start off blue, using less mana can't hurt.