Is it absoulety necessary to leave a 3/4" gap between the H.W.F. and the wall? Most of the installation instructions I've found on the web state that I should leave a 3/4" gap. That will force me to either cut the drywall back or install quarter-round or similar since my baseboard is only 1/2" thick. Any other suggestions?

Your baseboard is most likely 9/16"; most are today. A 1/2" gap is sufficient in most circumstances.

Is a 1/2" gap sufficient given that I am laying 3/4" BR111 Braz cherry in a room that is roughly 16x19?

I think so. But if you want, undercut the sheetrock up about 1" from the subfloor and this will gain you another 1/2" so you will only need a 1/4" gap between the walls and the floor IF you do this.

Thanks again Gary, you're absolutely correct, I remeasured my baseboard trim and it is 9/16" thick.