I emailed Google and got a few clues from them:
- Google does not track software incompatibilities, so there is no way to know if TotalFinder is responsible for the random " (1)" file duplications. I suspect it's not.
- Google Drive does not support resource forks or aliases. Dropbox has the same limitation, but it does not destroy the files like Google Drive does.
I have briefly documented the issues here.
My final word is that Dropbox is safer to use than Google Drive, and that both are limited in such a way that they cannot act as perfect substitutes to external hard drive backups.
As you've noticed, recent versions of OS X only display generic folder icons in the sidebar. So why does Dropbox get special treatment?
The short answer is that Dropbox uses undocumented API to accomplish this. In non-technical speak, it's a special hack that's installed by the Dropbox application.
Some curious folks on StackOverflow found the specific mechanism used by Dropbox: good ol' mach_inject. The same bundle provides both the toolbar item and the sidebar icon.
If you're curious, the resource files live here (at least, on my system):
/Library/DropboxHelperTools/Dropbox_u502/DropboxBundle.bundle/Contents/Resources
Clearly, Dropbox goes to great lengths in order to integrate seamlessly into the Finder. Lacking similar treatment, other ordinary folders have only generic icons in the sidebar.
Edit 10/1/2015: As of Yosemite (10.10), there is a new, sanctioned Finder Sync API for integration, as doovers points out in another answer. In El Capitan (10.11), System Integrity Protection will not permit the old mach_inject-style approach.
Best Answer
Sadly, this is simply not possible in macOS. The Google Drive app, and now its successor Backup and Sync from Google, give the actual folder named "Google Drive" its custom icon in the conventional way, much the same way you could paste a custom icon on any other folder.
Unfortunately, all custom folder icons are ignored when displayed as Favorites in the Finder sidebar. You can confirm this by adding any folder with a custom icon to the sidebar. You can also show your home in List view to see that "Google Drive" does indeed have a custom icon optimized for that tiny display size. Finder is simply designed to ignore it, presumably to keep the Finder user interface as uniform as possible.