How long can you work on the laptop with power plugged into it?  Until you
give out or some component of the computer itself gives out.

You can always work with it plugged in and the battery fully charged.  It is
not necessary to unplug it once it is charged.

However - it is good to let you know now that if you plan on using it
plugged in *a lot* - you are likely damaging the battery with heat by
leaving it plugged in after it is fully charged.  This (among other
reasons - mostly because batteries just don't last as long) is why most
laptop manufacturers - no matter the length of the warranty you get on the
laptop - only warranty the battery for approximately one year from date of
purchase.  My suggestion - if your laptop will allow it - if you plan on
using it plugged in most of the time - remove the battery.  Make sure to put
in the battery an hour or so before you will be taking a trip with it that
will require battery use and charge the battery for good measure.