Okay, My Roan Appaloosa Mare(Varnish/Frosted, what ever you want to call her) Is most likely bred(we know she is preggo, but don't know the stud) to a Dun Quarter Horse Stud, who I am 99% sure is Homozygous Dun looking at his past foals, I am not sure if he is a Silver or Agouti carrier, but anyway.
First off, you can't tell if a horse is homozygous for dun by looking at their offspring. Well you theoretically could, but it would require them to have over a hundred get, without a none dun at all.
Considering you use the term "dun" and not red dun or grullo, I am going to assume that he is a bay dun. If his base colour is different, feel free to correct me. If he is bay dun, he has at least one black gene, and at least one agouti gene. He may or may not carry red. Without testing, I would assume he is hetero for dun, black and agouti. I would also assume he is not silver, since silver is fairly uncommon in QHs and is in specific lines of the breed.
I was trying to figure out if Appaloosa patterns(Leopard Genes) are random occurrences or if they are passed down from parents to foal.
The reason I question this is I have seen people breed Leopard pattern studs to solid mares and get blanket patterns, or seen people breed Solid studs who carry the Leopard gene and get Blankets, Leopards, and Snowflakes.
Like since she is a varnish roan(with what seems to be like a few stockings and a blaze) is the foal going to most likely be a Varnish pattern or could it also be another pattern such as leopard, blanket, snowflake, etc?
Colour patterns are always inherited from the parents of the foal. Always. As for the intricacies of LP and PATN inheritance, well that is not my thing and I have no way to explain it so you will understand, as I barely get it myself. Someone else will surely be along that can.