Zeppelin,
(a) The battery acts as a UPS
(b) Depends on the battery type, some are meant to be kept at a reasonably high charge level rather than being allowed to discharge fully before recharging
(c) The old adage applies that one should only remove batteries from things if you are putting them into storage.
(d) I've got a 2009 vintage Apple laptop with a non removable battery, spends most of its time plugged into mains.... battery still gives its full rated 3.5 hour usage time when off mains.