Typically, bears and mountain lions are going to be long gone before you know they are there- my horse will let me know if he smells something, and remains alert, but no biggie. In grizzly country, the universal rule as I was taught is- it’s the bear’s trail, let him/her have it when necessary! But the only time I would be concerned is being inadvertantly between a momma and her cubs. If in heavy woods or willows, it’s often prudent to just make lots of noise so they know you are coming. I carried bear spray on my recent treks in Wyoming and Montana- on my person- and others had guns, but truthfully I would not consider using either one from the back of the horse- when I would need them is if I am on the ground, separated from the horse, and facing an angry bear. Then, if using pepper spray, I just hope I am upwind from the bear otherwise we’ll all be rolling around on the ground screaming in pain!
However, hmmm, putting it delicately…mares in heat, as well as, er, humans in heat, ARE an attractant for both bears and cougars, they can smell blood from miles away- something to keep in mind.
As for snakes- can’t say how many times my horses have stepped on or over one over the decades, rattlers, moccasins, copperheads, never a concern. I do avoid certain trails, where I know rattlers might be sunning themselves say at elbow level- why take chances? But generally they are going to be more active in the transitional am and pm hours- most of the day, they are in the shade, keeping as still as possible, they do get quickly stressed by moving in the heat. The one time I was on a horse and a rattlesnake was noticed, in a shady spot, and commenced to rattling (just before the cowboys killed it)- my horse definitely took notice and stayed a respectful distance away from the offending noise.