It depends on the breed. Like with Aussies, there is a short-haired gene but it's been so carefully selected AWAY from that you are pretty much never going to see a short-haired Aussie.
In some breeds, "with white" is dominant, some it's recessive, some it depends on the lines and the genetic makeup of the parents. I've seen dogs with minimal "flash" throw pups with a good amount of white to them.
You can fairly well predict the colors of the pups if you know exactly what the parents are carrying and what's behind them. I'll use Aussies since that's what I'm most familiar with. If one parents is say, homozygous for red and the other parents is carrying at least one red gene, odds are the pups are going to come out all red. But if you've got a set of parents that are heterozygous for both red and black (regardless of merle, though that will show up in some of the pups if one parents is merle) you'll have a mix of colors and markings.