I do think a lot of this is for dramatic and narrative purposes, particularly in the most recent season of Game of Thrones, but you do see this throughout the series as well!

Disclaimer: I specialize in the 12th Century. Game of Thrones is based loosely on the Wars of the Roses which took place in the 15th Century, so warfare had advanced a bit more.

It is certainly true that Aristocrats would have been better trained and equipped for combat than regular foot soldiers. These footsoldiers would have been lightly armored, and used what weapons and equipment were available, and what their liege lord was wealthy enough and generous enough to provide. It may also depend a little bit on the Lord's interest in warfare and their reasons for pursuing warfare. This is why you see the Lannister troops uniformly dressed, and Stark bannermen dressed consistently, but often mis-matched. Even then, the armoring in Game of Thrones is a little too consistent to really be a realistic depiction of history, but does a good job of letting the viewer know what belongs to whom.

In terms of weaponry, the Aristocracy was much more likely to use weapons specifically designed for combat, such as swords, maces, and lances. Legendary, named swords are a huge theme in medieval literature, i.e. 'Excalibur' in King Arthur and 'Durendal' in the Song of Roland. Richard I actually claimed to possess Excalibur at one point. This is comparable to Valyrian Steel swords all having names Ice/Oathkeeper/Longclaw/Etc.

Everyday footsoldiers, who until the 16th Century largely were not full-time professional soldiers (mercenary companies notwithstanding), often used makeshift weapons or tools. This is how weapons like axes, clubs, hammers, and polearms like halberds came into use. These easily double as tools or farm equipment in peacetime. Young men of all classes would probably have had some knowledge of archery, an essential skill for hunting. Bows were relatively easy to make compared to metal weapons and could be fashioned at home with gathered materials. Ramsay is probably a bit unusual in his accuracy, but he is also depicted as an avid hunter, if an unorthodox one.

Curiously, a lot of historical aristocrats also seem to have 'plot armour' of a sort. This was because in the Middle Ages, nobles did not often kill each other in combat. Sure, it happened, but it was actually considered unchivalrous and dishonourable. It was far more common for medieval aristocrats to take each other 'hostage' after disarming or incapacitating each other in battle. Killing a fellow aristocrat would have invited an escalation of conflict as the family of the injured party would likely seek revenge, and may be able to then appeal to a higher power, like a King or other overlord to intervene and administer justice. However, taking each other hostage: totally acceptable, as long as your terms are reasonable.

This is why in Season 1, Robb Stark takes Jaime, and other Lannisters, hostage, rather than killing them, and why he is forced to execute the Karstarks who murdered the hostage Lannisters. They have not only undermined Robb's authority, but have taken away a valuable bargaining chip that he could have used to financial, territorial, or political gain. Unfortunately, Robb was caught between a rock and a hard place, as the Karstarks also wanted revenge for their own dead. Unfortunately this kind of falls apart when the characters have this plot armour when fighting the White Walkers... Who knows, maybe the Night King needed cash?

You're also absolutely right about the Cleganes. They have reached their position from Patronage. Now, this did not mean necessarily that a Lord would choose any peasant who happened to be good at fighting to become a noble. Unfortunately, people were quite snobbish about social class, and raising a family from peasants to nobles would have taken several generations. It was typically more common for lower-ranking aristocrats, younger sons of nobles to seek their fortune through combat skill if they were not destined for the Church. The most famous example is William Marshal, a younger son of John Marshal, who had essentially held the post of horsekeeper and head of household troops, had no land of his own to inherit. Instead, he eventually caught the attention of Eleanor of Aquitaine and her sons, who later gave him lands and titles in exchange for his years of loyal service. As he would have had no income initially, he would have relied on them to supply him with weapons, armor, horses, and room and board. Similarly, the Cleganes started as kennel-masters to the Lannisters, and were eventually given a Lordship.

Sources:

Chivalry, by Maurice Keen.

Medieval Warfare: A History, by Maurice Keen

War in the Middle Ages, by Philippe Contamine

and anything written by Matthew Strickland.

Unfortunately, the question is somewhat impossible to answer. We don't have a benchmark for what can be considered the average soldier or the average nobleman. And even if we did, that average would change over time and place. Medieval Europe is a rather expansive locale.

Now, that all said, you are on the right track that a trained nobleman does have some serious advantages to allow them to increase their skill, or at least survival more than a common soldier.

Noblemen such as Fiore, Boucicaut, and William Marshal certainly have stories about them being much greater warriors than anyone they faced. Each of whom have at least one story of them facing multiple opponents victoriously. Fiore (an unindentured nobleman from the mid 14th century) claims to have had enough wealth to travel across the Holy Roman Empire and Italy training with several masters to increase his skill. A feat that would be largely impossible for most of the common soldiers. In his book Flower of Battle, where he gives some advice on fighting, he discusses his time as a tax collector and how he fought against and beat several ruffians. And he even gives a warning in part of his instruction on fighting to not teach the common people specific moves, because you don't want them to learn how to fight well enough to confront a member of the knightly class.

Similarly, Boucicaut (Jean Le Maingre a French knight form around the same time as Fiore) in his biography discusses his rigorous training regimen to make certain he is always in the peak physical condition. Involving climbing, jumping, dancing, riding, and somersaulting in armor for most the day. It is hard to imagine that a common soldier would have the time to keep up Boucicaut's training, especially since many were not professional (again depending on location and time).

And we can get to the truly ridiculous stories of military prowess. There's William Marshal who was winning tournaments (which were more like mass brawls) and leading charges when he was in his 70s. And of course "Le bon chevalier." Pierre Terrail the French knight, fought at the battle of Garigliano in 1503. Supposedly, he held the bridge, alone, against 200 Spanish soldiers which allowed the French force to retreat without being completely routed (temporarily anyway). And he survived and became famous for it in his own lifetime. So, while it is probably safe to assume that the exact number of Spaniards he fought was embellished, nevertheless, the idea of a single well trained knight fighting against many wasn't unheard of within medieval and early modern military history.

There is of course also the matter of equipment. Simply put, the best armor of the time can take a lot of punishment. There's a reason people wore multiple layers of padding and metal that surrounded their body just to survive fighting in a big melee. No matter how good you are in a fight, when facing multiple opponents you can't keep tabs on everything. You're going to get hit, that's inevitable. But good armor will basically negate being hit at all where it covers (I'm simplifying a bit). And the best, most expensive armor covered more completely and was better made than whatever most soldiers were wearing. Though over time, the gap between your regularly infantryman to your man-at-arms shrunk a bit as equipment like massed produced munitions plate became available. And those soldiers who did decide to become professional would invest in their armor over time for good reason. For an example you can look up John Hawkwood, a medieval mercenary who started as a longbowman then became a professional mercenary and by the end of his life was a famed (or at least infamous) condotteri with propaganda images of himself in plate. But where it took Hawkwood some time to reach that prominence, a wealthy member of the knightly class simply starts with similar equipment.

The other reason we get moments where nobles survive the impossible is because it was far more profitable to take them alive. After a battle captured knights were round up and either used as political prisoners and bargaining chips, or ransomed off for wealth, by those who captured them. There are accounts of groups of soldiers being able to take a knight captive, and how they would handle that situation. They would give the knight to their lord or commander (who had better capability to house and protect the captive than the soldiers) for cash either directly. So they wouldn't want to kill a knight if they could help it.

Now all that said, there is of course some form of plot armor going on with the characters in GoT. Ramsay charges into a melee shirtless and doesn't receive a single scratch. No one wears their helmets. The armor designs is mostly to look good. The tactics are simplified to the point of ridiculousness. Jaime doesn't know how to fight, even before he lost his hand. Wounds that in the books start as scratches that then get infected and can kill off or at least incapacitate the best of warriors get brushed aside in the show. Now none of this makes the show good or bad, by itself.