who makes the choice to cancel flights?

Air traffic control usually does.  (I guess that's the 'airport' in your options.)  Individual airlines don't have resources or experts dealing with this.   

Depends on the reason behind the flight cancellation. For example, if it is due to bad weather, airline companies use APIs from service providers like Climacell to judge whether it is safe to fly. Of course, air-traffic control would also have this information through NOAA weather data. This is applicable to America but I'm sure the system in other countries would be similar.

It's on a case by case basis.  Usually the pilot in command decides if the weather is outside the allowable minima for flying or the aircraft is not fit to fly due to a mechanical issue.  

However, sometimes the airport operator may also decide to close the airport.  For example, the Port Authority of New York and New Jersey has closed the JFK Airport when the snow blocked the Van Wyck Expressway, the main route in and out of the airport, even though the runways were clear of snow.  This is so that ambulances and fire trucks can get to the airport to assist the airport's own Fire and Rescue in case of a crash.