I'm a rider from way back when. Maybe I can shed some light.
"Head of the gallows tree" implies hammer-headed and thin-necked, with a high head carriage that would keep the bit from being effective -- all signs of ill-breeding and bad temper. In conjunction with the dun coloring and other attributes, this horse is a capable outlaw, with a hard/insensitive mouth and an evil temper. Slugging means leaning on the bit, either running away, or in exhaustion. I think the latter in this case, from context.
Snaffles are a jointed bit, two rings or dees connected by a jointed straight metal piece that sits in the horse's mouth. It works by flexing at that joint in the center of the horse's mouth, thus pinching the jaws and corners of the mouth. The reins attach directly to the rings/dees,as does the bridle, and normally there is no extension piece on a riding bridle; snaffles on driving bridles can have an extension down from the rings to keep the reins from going over the horse's head if he tosses his head. It tends to be a pretty mild bit, though there are variations that can make it pretty intense, such as making the straight pieces thin or out of twisted metal. The "bars" are thus the jointed piece in the mouth, and playing with it implies that the horse is light-mouthed, easy to manage and responsive -- or not nearly worn out yet.
Does this help?