I tell my clients who never unplug their laptops to remove the battery, and when they need to move the laptop plug in the battery the day before to make sure it's all ready. I don't know if they have fixed it but the laptop battery will drain a little and the charger will top it off, repeated cycles of this can wear a battery faster then only using it when needed.
I don't believe this is correct, at least if you mean that the laptop drains the battery then the charger re-charges it. Power goes directly from the charger to the electronics, not via the battery. If you mean self-discharge, maybe.
From what I've heard, the main reason a battery could last longer outside the laptop is heat.