How easy is it to run coax cable with the given configuration?
You would just need a jumper from the COAX plate to the new box next to it. Or you could just run a COAX through a hole in the blank plate then up to the TV.

Question is, WHY do you need COAX at the TV? Do you not have/need a cable or sat box? 

Typically for new TVs we just run one or more HDMI cables form the cable box/blue ray/game console/etc up to the TV.

You are lucky to have that sleeve from the floor area up to the TV.

(And my post is fine. Writing "COAX" does NOT come off as shouting. That's just silly.)