The palomino stud is ee (homozygous for red) Since the black horse had a chestnut parent, you know she's Ee (heterozygous for black/red) So you have a 75% chance of getting a red based foal.
In the event that you get a black based foal, it's possible for it to be black (no agouti) or bay/brown. The mare, being black, doesn't carry any agouti. The agouti would have to come from the stud (doesn't show up on red based coats, so he can carry it without any outward signs). If he's homozygous for agouti, then the black based foal would always be bay or brown; if he's heterozygous it's 50/50 whether a black based foal would be black or bay/brown, and if he doesn't carry any agouti, the black based foal would be black.
Then there's a 50/50 chance of the cream gene being passed on. (BTW, the others are correct that the stud's sire or dam must have carried cream for him to have it... could be that the chestnut dam was a very dark pali mistaken for flaxen chestnut, or the sire was a smoky brown that didn't show the cream very well) The stud looks pali to me, but always possible that he's actually flaxen chestnut as well.
As for the appy patterns... I'm not advanced enough to figure those out. With 2 appies you could get a solid (no appy genes passed on) to a few spot, or anything in between.
ETA- Just noticed that the mare's dam was pali... which means she could be smoky black (hiding a cream gene) so you could possibly even get a double dilute.