Does using a macbook with an external monitor damage the health of the battery? I have an (oldish) MacBook Pro that I use with an external monitor, which requires that the MBP power adapter be plugged in all the time. Does that damage the health of the battery? I ask because my battery, replaced about a year ago, will now hold almost no charge after being used almost exclusively with the external monitor during working hours and plugged in all that time.
While the external monitor itself is not the cause of the 'weakened' battery, leaving your MacBook Pro on AC power all of the time will definitely affect the battery's performance.  It is essentially under a constant state of charging, even though this is regulated by the charging circuit, which will eventually cause the battery's chemistry to begin to break down.

Current batteries will typically last the longest when used regularly, but not completely discharged with each use.