1- Counter canter is actually a dressage exercise, and it's okay if you're doing it by asking your horse to actually pick the outside lead or just by doing a diagonal on canter to from the right to the left and keep on cantering in the right leg. In conclusion you should do it, only when asking your horse to do it. And if it isn't your intention to counter canter, and you've asked your horse to canter and he picks the wrong lead, you should immediately ask your horse to trot again, or walk, if you're doing a walk to canter trasition, and give him more correct aids so he picks up the right lead.
If your intention is to counter canter, make sure you don't push the horse too deep in the corners, so he can keep his balance and doesn't break into trot who does a flying change.
And for your second question, it's a chestnut as it was said before.