The simple answer is that there is no point at which dragoons were first employed because Dragoons as a kind of unit evolved slowly. They were simply a natural answer to the need to move infantry very quickly, something that has happened now and then for a long time. Some time in the early 16th century the first real regiments of mounted infantry armed with firearms were raised (although exactly were is debated).

These regiments were used more for their strategic mobility than tactical mobility, that is, horses were generally kept rather far from the battle, the value was in being able to move an entire regiment much further than your enemy, especially so in an age where warfare was largely about strategic positioning.

As dragoon regiments became more common their roles evolved. They were commonly employed in situations where you needed to get troops in position fast or be able to withdraw fast. Typically this would include quickly moving in to secure an important location (such as a bridge, a mountain pass or a crossroads) and holding it until the main body of the army arrived. Other uses for their mobility included patrolling much larger areas than a normal regiment of foot could have managed, establishing or holding distant outposts or simply maintaining lines of communication.

As we see here dragoons were relatively rarely sent directly into combat on horseback early on (as they were at a severe disadvantage against the better trained and equipped cavalry) and infantry (as they usually lacked numbers and training to take on a regular infantry formation unsupported). In battle dragoons often held a position while awaiting reinforcements or acted as fire support to another unit, for example cavalry who they were able to keep pace with, unlike infantry and artillery.

Aas time progressed the dragoons generally sought to rise in status, and generally (Britain excluded) succeeded. The term dragoon became increasingly synonymous with a conventional cavalry regiment in most armies. By the time of the Napoleonic wars the dragoons were simply cavalry, not as heavily armored as the cuirassiers, not as mobile as the light horse.

By the early 20th century cavalry was falling out of fashion and mostly came to be mounted infantry again (with a few exceptions), thus basically coming full circle, from infantry on horseback to cavalry to infantry again.

To answer your question about their horses, it depends on which era and army you're asking about. Sometimes they were left far from the battlefield when the dragoons acted as conventional infantry in battle, sometimes they were left a short distance back when they acted as tactically mobile forces in support of another unit and sometimes the dragoons rode them into battle when acting as cavarly.