painting a cork floor (previously coated with polyurethane varnish)
I honestly do not think painting a floor would be a good idea. A solid color paint or stain will likely chip and wear unevenly. As in wherever your high traffic areas are will look awful in no time at all. 

If you are to paint it you will need to sand the gloss of the previous coating. This glass will prevent any other paints or stains from correctly adhering to the surface. The same exact rule applies to painting a glossy wall or any other surface.