There are two ways to use Dragoons, as a semi-heavy cavalry (the French during the Napoleonic Wars classified them as Medium cavalry) or as dismounted infantry.

In the cavalry aspect, they were given straight swords with brass helmets. While armed with a cavalry musket/carbine, they would be used as heavy cavalry. Heavy cavalry would be used to push or break an enemy line and cause a break through in the enemy's line. From there, infantry would normally push forward and pour through the gap.

The second way Dragoons were used was as mounted infantry. By mounted infantry, they mount onto horses, go to a location, dismount and then move forward to skirmish or fight as a line. The reason they fought dismounted was for ease of reloading (reloading anything on a horse is difficult because it is a living animal) and because horses have to be very well trained to handle the sounds of battle, let alone someone firing a musket on top of them.

Generally, it's because it's very impractical for a dragoon to be on a horse. Further, there wasn't a need for dragoons to be mobile as they aren't dedicated horse skirmishers but again as mounted infantry. Even as late as 1800 there was discussion on whether they were mounted infantry or mounted infantry that seemed to never be resolved with certain.

Edit: If you want to look for your question about crossbows and early guns, you might want to look at our FAQ which has several answers on the topic

Right off the bat I need to address something that is seeming glaringly apparent and that is what seems to be rivet counting. That is, trying to look at the very micro level of military history and apply 'stats' to things or play the game of "well this weapon has twice the firing rate of this weapon, clearly these people were illogical for not using it!" or something to that effect. This isn't a game of Total War or RISK and I mean that in the kindest way because this is a mindset which I believe must be stamped out to get any real understanding of military history. You can't think of it as stats in a video game. It's easily one of the most common things people fall into when they're new to military history but it's also one of the most dangerous and toxic mindsets to have in any study of it. Now to proceed:

What happened to the javelin? Why don't medieval army foot soldiers carry 1-2 javelins to throw before an engagement like armies of antiquity did? I found this answer[1] , but it doesn't quite answer why it lost popularity. The answer stated that it didn't, but it clearly did lose popularity as depictions of medieval warfare lacked the pervasive javelins of earlier times. Even if the argument is that the bow replaced the function of the javelin, wouldn't it be advantageous to have 100 bowmen and 500 javelin-equipped footsoldiers over 100 bowmen and 500 sitting ducks for enemy ranged weapons?

But your question creates a false dilemma. The issue is not between "100 bowmen and 500 javelinmen + soldiers or 500 bowmen and 100 javelinmen." The fact is is that the bow wasn't the only skirmishing tool available. Javelins, slings, crossbows, bows, and whatever else would be used throughout the entire period. The fact is is that every skirmishing tool had its advantage and disadvantage and they were all used appropriately. The Romans used bows, javelins, slings, and whatever else as well depending on the environment.

What you're talking about is the Roman tradition to use the Pilum, a much heavier option to a traditional javelin carried by 'regular' infantry, to soften the enemy before making contact; they were not acting as skirmishing bodies. Like I said, the Romans had dedicated skirmishing bodies like slingers, javelinmen, and bowmen alike who performed screening duties depending on where or who they were fighting. The issue of why didn't Medieval skirmishing bodies use javelins as much as, say, bows was explained in the very post you linked. So then why didn't regular infantry bodies carry a javelin-like weapon like the Romans? Because they weren't there to perform skirmishing duties and they were not equipped, trained, or there to act in any way that necessitated something akin to the Roman Pilum to "soften" the enemy before contact. Especially considering they were, generally, carrying massive pikes the later in time you travel.

Another important factor is it is proceeding the fall of the Roman Empire and thus massive fracturing of states -- warfare has been reduced from a centralized aspect of a massive, organized state to something a bunch of barons and dukes and princes perform against each other. It has become much smaller scale. Combine this with the fact that the vast number of 'regular infantry' in this period were either the dregs of society forced into conscription or peasantry obligated to conscription -- both largely or entirely untrained -- and a picture becomes pretty clear. They were not organized and professionalized military bodies like with the Romans and thus could not operate under the same level of sophistication. They also lacked the level of financial and social backing that the Romans could so they lacked the same level of material backing.

How did the arquebus and matchlock musket ever replace the crossbow? The crossbow fired with greater range and accuracy. The matchlock musket needed you to keep a slow-burning match constantly lit. Both needed very little training.

The crossbow didn't go "boom" and create a shit ton of smoke. That sounds silly but it really is the reason. War is psychology first and foremost and this goes right back to my first paragraph about rivet counting. It doesn't matter that bows or crossbows have higher "firing rates" than early muskets -- a musket is straight up loud and goes boom while also creating a shit ton of smoke. Now imagine 100 or 200 going off at once in your general direction...and then you can't see anything anymore and your ears are practically blown out...and 30 or 40 of your buddies slump down next to you. That's psychological impact that no other weapon before could match.

With that said though it wasn't some overnight change. It took the entirety of the 16th and 17th centuries before they began to become mainstream weapons for a reason and it's like you said -- the technology was not refined. The earliest use of musketry were by Spanish Caracoles which were mounted screening forces in the 16th century. Basically it would be groups of cavalry armed with two pistols who would run up to the enemy formation, break at a 90 degree angle, fire off both shots, round back and reload, and then go up to the front and do it again. This could not be formed with any other form of weaponry besides the bow and that required a lifetime of experience and training while two pistols required a few months at most. Gradually though more hefty weapons would make its way into infantry formations in smaller numbers. Seeing just a couple dozen muskets for a few hundred men was not uncommon as even the psychological impact of those could turn the tide.

Throughout this period though in the Tercios (dense, mixed formations of various weapons all supporting each other on the most basic level) and formations like it would use an absolute smorgasbord of weaponry between swords, pikes, halberds, crossbows, bows, and probably whatever else you can think of. Over time, particularly the 30 Years' War...which is coincidentally the time when the Flintlock musket began to come into play in Europe which, largely, removed matchlock and its more primitive firing system. Through this time the psychological impact of the musket described above along with generally more penetrating power would, by the end of the 30 Years' War, create a situation where infantry formations would run off the tradition of a 1:1 ratio of pikemen and musketmen who would protect and support eachother.

Why did dragoons ever dismount? If they bring their horses to a stop they could easily discharge their weapons with a steady hand. Also, before rifled barrels the accuracy of guns were atrocious anyways, so why bother to dismount to fire their guns when they could have the advantage of being more mobile? This one really confuses me, as I envision men using guns on horseback like the mongols used bows, but apparently the dragoons had to be on foot to use their guns? This does not make sense to me.

/u/DonaldFDraper goes into good description of it but let me try to clarify something he left out -- this is another case of you conflating two things like you did in your first question. You're treating dragoons as if they're supposed to be acting as a skirmishing body when they're not. They're acting as mounted infantry -- they were using their horses to maneuver quickly across the battlefield and flank the enemy. However you can not put your horses in a dense line formation and then have them stand still right next to each other and engage the enemy in combat. They did have those who fired guns from horseback but they were more screening forces rather than infantry designed to actually decisively engage the enemy. That could only be meaningfully done on foot. Think of the horses less as battle tools and more as like a motorcycle or a humvee for a modern example -- it moves the infantry around quickly so they can outrun the enemy counter-maneuvers who are on foot.

Whatever happened to War Elephants in the west? I know in India and Southeast Asia they continued to be used, but what change caused their decline in the west? Carthage used War Elephants, and so did the Persian Empire before them. The Hellenistic Kingdoms that succeeded the Persian Empire used them. Then you never hear about them in Middle-East conflicts anymore. The crusaders never encountered war elephants fighting arabs in the holy land. Why did arabs discontinue elephant warfare? I found one record of one elephant "Abul-Abbas", but he was more ceremonial than a beast of war.

They went extinct. Simply as that.

During the Bronze Age, there were elephants in North Africa, southern Egypt, the Fertile Crescent, southern Iran and China up to the Yellow River. By the beginning of the Middle Ages, they had died out in all of these regions, and only existed as extremely expensive imports, mostly from India. Since war elephants are only effective in groups (and even then, can be easily scared away and become a danger to their own side in battle), there came a time when not even the richest emperors outside of the elephants natural range could afford a troop of elephants and their handlers (who, by the way, had to be trained themselves and were hired overseas along with the elephants themselves). So for the West, elephants ceased to be a military unit, and became a prestige thing. Something you could own, show off and offer as a diplomatic gift. And since you can show off a single elephant as well as show off a troop of 10 or 30 elephants, what's the point in paying for more than one of the increasingly expensive animals?