1. How do I get my machine to recognize my device via USB? Again, I'm on a Mac running OS X 10.8 Mountain Lion on the Verizon's version of the Galaxy S3 which currently has ICS.
Try Android file transfer with TCP again
2. What's the best method of syncing music and playlists, either via USB or wireless?
Kies is a pretty good contender...you can download the kies program (works like itunes basically) , or even use it in your browser wirelessly by opening up Kies Air, an application that comes pre loaded on your phone.
3. Is there a way to root the device using a Mac and if so, is it possible to get a clean copy of Jelly Bean (or another recommended image?) that will still work with the Verizon "necessities"?
I believe there is a one click root solution for the mac, But your best bet would be to get a computer that runs windows, dual boot windows on your mac, or run windows in parallels to root your device that way. There is a page on XDA forums that lists all the Stock image files for your phone, including some ones that get rid of bloat, or come rooted already.
4. Finally, I did a full charge last night (phone has been on battery for about 3 hours now) but the battery is already down to 84% (with very little usage) - what are some techniques you guys use to conserve?
1. Disable apps that you dont use (mostly bloatware) by going into settings-application manager-all- and selecting the ones you do not use.
2. Set brightness to auto in brightness settings
3.Turn Sync off if you aren't using it by going into settings-accounts and sync- and turning it off via the switch on the top right of the screen.
4. Getting a black wallpaper...This tip dosent save much battery, but it does save a little
5. turning bluetooth and wifi off when youre not using it...those are big battery suckers
6.Turning off NFC, WIFI direct and S-Beam When youre not using them by going into settings-more settings and switching them off
7. Turning motion activation off by going into settings- motion activation and switching it off
8. Using location services only when you need them (GPS, VZW Location Services, Google Location Services) By going into settings- location services
9. If youre not in a 4G covered area, get an app called "Phone Info" from the Play Store and open it, scroll down until you see something that says "LTE/CDMA/EvDo" and click it. Turn this setting to "CDMA auto (PRL)" This basically turns of the 4G radio so it isnt looking for a 4G signal where there isnt one. To reverse this, just turn this setting back to "LTE/CDMA/EvDo".
10. Turn off wake up command. Double press the home key until you see "S Voice" pop up. press the menu key (the button next to the physical home button) and press settings, and scroll down until you see wake up command and turn that off.
11. Touch key lights.. Limit the amount of time they are on by going into settings- display and pressing "Touch Key Light Duration" and setting it to a setting that you are comfortable with
You dont have to do all of these things... but try to mix and match the ones that suit your personal usage
Hope this helps!