How can I hide the gap at the base of the stairs and the door, when installing laminate flooring?
Typically the solution at the bottom of the stairs is to use a quarter-round (Usually 3/4" thick) to cover the flooring and the bottom of the carpet.  

You'll need some long nails to get through the carpet, so it's best to pre-drill the nail holes so you don't split the trim. 

This is the best image I could find.  It lacks carpet, but will demonstrate the basic idea.