If I have several functions located in columns throughout my
spreadsheet, is it possible to make them continue downwards, so that
whenever a user adds a new row, whether it be the 15th or the 1500th,
those formulas will still be present.

One way in Excel 2003 is to use
Data List Create list


Also, along those same lines, is
it possible that when a user enters data into a new row, I can make it
so that borders will automatically appear?

Try conditional formatting, as another responder has already suggested.