The Amiga file structure has a time/date stamp embedded within for file creation and file last modified if I remember correctly
Correct
when checking your harddrive for errors etc, you do not want to be sitting there pressing ok to requesters hundreds of times just because the check says 'file set to future' of some such report from the likes of Amiback tools or Quaterback, maybe even good ol' Dave Haynie diskSalv
Naw... It's only a warning. There's no harm in having a wrong date on a file. And there's a box to turn off the "improper file date" warning, anyhow. And, really, you probably don't want to be using those tools except in very controlled circumstances anymore, anyhow. Things like SFS have (thankfully) made them redundant.
If you wish to keep your amiga file system all happy and smiling then you do require a battery
More Amigas were originally sold without batteries (A500,600,1200) than with (A2000,3000,4000). There doesn't seem to be a rash of problems on those machines. (shrug)
The A3000 is a slight exception, as it also uses the battery to store some information about the SCSI configuration of the machine. (Though under most circumstances you want to run in the default configuration, anyhow...)
All-in-all, the replacing the battery is quite optional. The important thing is that you get the old one out before it leaks. :-)