Aficat is right about how genetics work. Only it's not the genes that are in pairs, a gene has 2 parts that are called alleles, one from the mother and one from the father. A pair of alleles forms a gene.
There isn't just one gene that gives a horse it's color, it's a combination of different genes that together make the color. So for example there isn't a single gene that makes a horse a 'bay-dun born gray', thats a combination of 4 different genes: black, agouti (turns black into bay), dun and gray.
The alleles of a gene can say YES or NO for a specific factor. The alleles of a gene can be different from each other, and in that case there is only one that shows. The one that shows is a DOMINANT factor, the one that doesn't show is a RECESSIVE factor. So a dominant factor only needs one YES allele to show, a recessive factor needs 2 of the same alleles to show.
Most factors are dominant, there are only a few that are recessive. Gray is dominant, so if a horse has one YES (G) allele for gray and is G/N it will be gray, but can also (50/50) pass one the not-gray allele.
And a horse isn't 'just' gray, all of the other genes are also present and can be passed on. So the mother of the horse in the first post didn't pass on her G allele, but did pass on other factors that she had.

And Aficat mentioned HERDA to illustrate recessive genes.

As for the roan foot, that's genetically the same as a white foot, only a minimal expression. Just like you have horses with just a few scattered white hairs on their forehead as a minimal exprssion of facial white.
She isn't gray, and won't turn gray. A gray starts to go white from the head, not from the hind end.