Pacing is genetic determined and therefore not unnatural for many horse breeds, like Icelanders, Lippizaners, Friesians, several warmbloods. Pacing used to be very popular for hundreds of years because of it's smooth ride. In the medieval ages pacing horses were the most expensive riding horses. Increasing use of coaches and better roads lead to different requirements in horses. After that pacing or gait horses disappeared gradually.
Tölt and pacing is not exactly the same. To make it visible look at the figures below.
4-beat Tölt:
(lateral) Pacing:
And this is the usual (diagonal) walk gait:
Not all gait horses are the same as HH said, so differences within the pattern are possible.
As for dressage we don't use the pacing or tölt, we use the basic 4 gaits: walk, trot, canter and gallop with several sub gaites, depending on dressage level, such as extended or working trot and gallop and several figures during the gaits, such as pirouettes, passages, appuyements etc.
There are even more gaites, such as slow gait, fox trot, rack, but I think this is enough for the time being.
Horse breeds have their own gait strengths and weaknesses. For instance our New Forests have strong trot, canter and gallop, but weak walk, they tend to dribble instead of walk. So we train them hard on a steady and relaxed walk.