Hello from new member with a Luke III

I recently purchased a Luke III HSS…excellent instrument!
Questions that were not (could not be) answered by the salesman at the music store:

1. What, precisely, does the 9 Volt battery do?
2. Is the battery being drained any time a guitar cable is plugged into the guitar?
3. Are the DiMarzio Transition pickups active or passive?
4. Should I get any buzzing at all when in switch position 3,4 and 5, (position 1 is the bridge pickup) a buzz that goes away when I touch metal on the guitar?

Many thanks,

RB