I think current US spelling is "gray" but things were greyer a while back.

I think old and gray go together - so it may be less a function of the natural color of the animal than it is a comment on its age.   Some how the "old brown mare" doesnt conjure up the same image.   Frankly I have no idea if horses gray with age - so this may be an anthropromorphism.

The term Yellow Dog Democrat is apparently a usage that significantly post dates the Civil War (if Wikipedia is accurate) but the term Yellow Dog as a disparaging comment most likely predates that. Interestingly Lincoln at one point was known for walking around Springfield with a large yellow dog. I have not been able to document if the Yellow Dog in the song was meant to refer in any way to Lincoln either negatively or positively - but I know people who believe that to be the case. On the other hand I have seen some indications that the Yellow Dog in the song predated or was totally independent of Lincoln - but I suppose even if that were the case some might have used the words retrospectively with him in mind.