I think horses are far down the chain of infection, cloven hooved animals like cattle and sheep are the usual animals to be infected. When there have been outbreaks, it's usually because an old burial pit on a farm has been disturbed, as it can live for hundreds of years. Tanneries used to be another place where it was easy to catch anthrax. I would therefore say that although not impossible, it is highly unlikely that you would catch anthrax from a horse skeleton unless it had been buried with cattle in an old pit.
That's just my understanding though, not saying it's ok!!
Depends on what the horse that owned the skeleton died of! If it died of anthrax, then yes. If it died of something else, no unless by an amazing coincidence it was buried with some anthrax infected cattle.