I'm not aware of anyone having bred lavender into their paints, but it could happen. And would require that BOTH parents provide the gene for it to show in their offspring.
Just as some styles and colours of clothing are more fashionable one season than another, so it goes with breeds and varieties. However, I haven't really noticed folks getting out of paints...
Khaki requires two copies of the dun gene. The off-coloured paints are not khaki. Khaki is a dilution of eumelanin (black pigment); the off colouring background in paints is pheomelanin (red pigment). A khaki paint would have pale khaki spots, just as a blue paint has blue spots.
First, there is no definitive answer as to whether there is or is not a separate paint allele of dominant white. Many of the paints have off-coloured hackles, and some are entirely off-coloured. My hypothesis is that gold and possibly autosomal red play into this off-colouring.
Those who...
Breed to a black or to a white from a paint breeding. Some people breed to regular, recessive white birds with good results. I am afraid that may come back to bite them in a generation or two.
Most folks that have bred paint to paint are unimpressed with the results.
Whether paint can hide or not is not unknown, although there are some who believe it can. Personally I am very skeptical of it hiding in a black bird; slightly less so of it hiding in a white bird. Is paint a separate, new gene? Or is it the same gene that produces exchequers? Is it produced...