I know some people swear by giving their new gadgets a full charge before touching it just after getting it out of the box. My question is, do you think I should give it a full charge while it is completely shut off, or would it be ok to use it while its getting its first charge?

I think you're OK to use it during the first charge. But, yes, many manufacturers recommend a full charge, followed by a full discharge around first use.

Even that may not calibrate the battery system though. A couple weeks ago I had to calibrate my OQO - which showed terrible battery life out of the box - and the battery life jumped from 1+ (improper) to 2+ hours (correct)

Let it charge for 20-30 minutes, then you may start using it while continue charging. Just to be on the safe side (you don't know the initial state of the battery - it may be very low for all you know).

The calibration cycle required on NiMH quick charger equipped items does not apply to Li cells. At the very worst, you might get a bad estimste of battery lifetime during the first cycle, if you don't do a dummy cycle - but I haven't observed that on the 770 or N800.