Chargers typically run at 14volts so no different electrically than an alternator being on, so wouldn't hurt anything, key on or off, just less charging with key on because system was drawing some of that power
Yes, disconnecting one battery cable to charge a battery is a good idea, chargers can fail and there are things like "power spikes" in home electrical systems, and that could ruin the battery, which is bad, but better than also taking out some of the vehicles electronics as well
Clicking relay and "no action" would usually mean bad connection for battery cables, remove and clean cable ends and battery terminals, BOTH cables
When an alternator fails it usually doesn't happen instantly, so it could have been failing for awhile and that can ruin the battery
So after new alternator is installed, test engine running voltage at battery, 14.0v to 14.8volts or so is OK, drive vehicle for a few days
After vehicle sits over night(4+ hours) but before starting, test battery voltage, 12.3v to 12.8v is OK, 12.2v or lower is a failing battery, replace it