Where does turbine vane and blade cooling air come from?
To build upon a few points in Peter's answer:

This diagram will probably help visualize it (I copied from this question How is the central hub / shaft casing of a two-spool jet engine assembled?): 



The seal between the disks is just a labyrinth seal.  Those chambers, however, are pressurized (by similar mechanisms as the cooling air was delivered to the blade). So there is a tiny positive flow of air out of that chambers, which prevents the hot combustion gases from flowing in.  Refering to the diagram, the space in front of the 1st stage disk is pressurized from air coming from the stg 10 compressor through the series of passages colored in orange.  The space between HPT stage 1 and stage 2 is pressurized by air the comes along the inner diameter of the HP rotor, from an impeller between stage 7 &amp; 8 compressor (in blue), and the space aft of the HPT stage 2 is pressurized from air the comes from the stage 4 compressor (by means of pipes on the outside of the engine, light blue color). 

Pretty much every cavity/chamber inside the engine is pressurized somehow, to keep air from going in undesired directions.

I don't know exactly how much air is used for cooling. It's on the order of 1% (i.e. I know it's less than 10%, but certainly more than 0.1%)