What are the printouts/paperwork a captain does at the check-in desk?
The paperwork he was looking at is the dispatch release.  This paperwork will have


Crew names
Fuel information
Filed route of flight
Alternates as needed
Current weather
Forecast weather at destination and alternates
NOTAMS for departure, destination, alternates, navaids and anything along the route of flight.
Deferred MEL items


There are also probably a few things I'm forgetting at the moment.

When the captain gets the release the first thing to do is make sure it is  for you -- correct crew names and airplane.  You'll check the route of flight and make sure it will work (weather) and evaluate the ordered fuel.  You'll review the destination weather, see if an alternate is needed and make sure a suitable one was filed.  You'll also check departure weather and make sure a takeoff alternate is specified if needed.  If the paperwork isn't right, you call your dispatcher, otherwise you sign it and take it down to the airplane.



The one-page item the captain got may have been a pre-departure clearance.  These are generally available via ACARS but can be printed if needed at a terminal.  They could also be other information such as ground delay programs or ground stop information that affects the flight.