Do horses have to obey traffic lights?

In some places, but not where I am right now, horses are given "prior right of way" because they were there before the motor vehicle existed. It's sort of like, "The pedestrian always has the right of way." The idea of "right-of-way" does not include the right of stupidity, even though it sometimes seems like that in Massachusetts. 

I have heard stories, perhaps from Amish areas, of some fellow getting drunk at a tavern, getting into his buggy or wagon, and going to sleep while the horse found its own way home. A horse in that situation will not be particularly careful about obeying stop signs or traffic signals, and it is thus up to the motorist to be aware of what might be on the roadway. But then, shouldn't we always be aware? Where I live now there are plenty of white-tail deer. They seem to have very little idea of handling motor traffic correctly, far less than dogs or horses. Encounters with deer are sudden and often brutal.

Ignore traffic controls, whether walking, riding, driving, or motoring, at your peril.

If the riders have any sense, they will observe a red light - or risk being injured/killed!   Go within a speed limit - irrelevant given how fast traffic can go.  Ride on the correct side of the road etc. - again if the rider has any sense, of course.   I'm not sure about 'has to'.

Yes of course. I’ve come across pedestrian crossing s where a bridleway crosses a busy road, and there are separate buttons set higher up for horse riders.   

Yes.  They are subject to traffic rules same as a car or bicycle.

Mounted riders are required to obey traffic signals and follow road rules, yes. Specific types of rules depend on the country and/or state that they are in.

They are required to obey all traffic laws IF they are ridden on surface streets.

Yes they do otherwise they could be hit by other traffic. 

Because every nation on Earth must have the same laws regarding riding horses on public roads. Are you mentally slow?