#2
I can't say for sure on the holy grail, but my two EHX pedals (a small clone and a double muff) both run on a nine volt battery.
#3
It takes 9v DC, and the EHX website doesn't say that it doesn't take a battery. It probably will. You get a 9v power supply with it, so why's it a problem?
#4
No it doesn't, power supply comes with it, I got a european one with mine though, which ****ed me off slightly...
Apparently it's 500mA so if it did run off batteries it would eat them pretty fast.