What part of the F-18's wing profile allows for high AoA flight?
You're correct that the leading edge extension (LEX) plays an important part in increasing the maximum angle of attack of the Hornet/Super Hornet. The LEX basically forms a vortex above the main wing, which improves the high angle of attack capabilities by delaying separation. See this question about vortex lift and also another related question. 

You can see the LEXs forming vortices over the wing in the following image. 



An F/A-18E Super Hornet assigned to Strike Fighter Squadron (VFA) 81 maneuvers over the Nimitz-class aircraft carrier USS Carl Vinson (CVN 70) during an air power demonstration. (U.S. Navy photo by Mass Communication Specialist 2nd Class James R. Evans/Released)


  Is high AoA associated with smaller wing width and high chord length?


There is no simple way to answer this question because what matters is not the aspect ratio of the wing per se, but the way the lift is created. In case the wings have high sweepback, the method of lift generation is by vortex lift, which operates in higher angle of attack compared to the normal (moderately swept) wings.

In other cases where the wing sweepback is less, the lift generation will be by conventional means and consequently, the high angle of attack performance will suffer, though this can be remedied by the use of canards, thrust vectoring etc.