What are these tiny, swarming, jumping bugs?
Very useful image updates! These are actually not arachnids but hexapods called springtails (order Collembola). Although springtails are often very tiny and hard to see without a lens, you happen to find some "larger" specimens. Specifically, these appear to be some species in the family Dicyrtomidae. 



 Credit: M. Bertone (source)

Your species resembles the image of Dicyrtomina minuta seen here or Dicyrtomina saundersi seen here (though I can't be certain of these source's accuracy).

In general, springtails usually feed on decaying matter, and -- you guessed it -- they're known for their ability to jump! 


  Springtails are so-called because they have a unique structure, the furcula, that allows them to jump for considerable distance relative to their tiny size. [source].




 Springing mechanism of a generalized springtail. Credit: Marianne Alleyne [Source] 

According to NC State Extension:


  Rarely springtails may become exceedingly abundant and may congregate in heaps several inches high on driveways, sidewalks and poolsides.


Collembola.org has many excellent close-up images as well as detailed descriptions of nuances and sexual dimorphism seen in the Dicyrtomina group of springtails.