Yes, that will work. Once you get backups working on the new TM drive, then I would recommend going back in the TM Preferences and removing the old TM disk from the TM disk list so it does not start nagging about not doing any backups to the old TM disk.
It isn't. When it works it is great. Multiple times I have tried to do restores and TM failed as it was corrupted. I would never rely on a TM as your only backup. Make sure you have at least 2 other backups - one via something like Carbon Copy Cloner, one in the cloud.