How can I remove calcium deposits from a drain?
Acid. Vinegar, Citric (aka sour salt), CLR, Barkeepers friend, phosphoric, etc. Keep applying (and re-applying) it, and see if perhaps you can get dry citric acid or barkeepers friend mixed to a "paste" or "slurry" form that might cling to the problem area - or fill the trap with acid and use sections of rope or rags (secured against going down the drain) to act as wicks. 

You might also want to install a softener if your water is so hard it's growing stalactites. Yeesh.