I'm answering my own questions in case anyone else has similar challenges with the 5D3. First, I'm not sure if use UBS-C to USB-A is what made the desktop icon correct (image of Drobo drive). Instead, it may be that when I restart the Mac it makes the drive a generic icon. Then I can unmount and mount it to get it to show the right drive icon.
I'm currently using the 5D3 with a USB-C to USB-A cable until Drobo releases the update to support me going back to the Thunderbolt connection. This USB connection is slower, but since the Drobo is just a backup drive, I don't care too much about that.
Now... The elusive problem with Drobo Dashboard not finding the drive (even the drive is fully functional in the OS). I tried everything. Uninstalled and reinstalled Drobo Dashboard (DD). Uninstalled DD and then manually removed any residual reference I could find to DD (ex. Full Disk Access under System Pref>Privacy). Nothing worked. However, I found a DroboTBT.kext file in Library/Staged Extensions/Library/Extensions. It took my a while to figure out how to delete that. The summary is: Disconnect Drobo from the Mac. Reboot to Recovery Mode. Use Terminal to execute 'csrutil disable'. Restart (normally). Delete the .kext file. Reboot to Recovery Mode. Terminal to execute 'csrutil enable'. Restart normal. Install Drobo Dashboard. Reconnect (for now, via USB). Open DD and it found the drive.