Hi there! So this is just a note for the mods and the OP, but some of the questions are framed as negatives ("Why didn't" someone do x/something happen). This isn't a problem as I am happy to answer these kinds of questions, but I don't know if there are definitive sources to frame these answers(medieval war customs, diplomatic customs, etc. are contextual), but if needed I can provide sources that I am using for context I provide. If there is something I am missing, please just let me know!

Anyway, on to your questions. So to start with, King John had surrendered on the battlefield. The background is that at the Battle of Poitiers, the French mounted a somewhat reckless charge against the English during which King John's personal retinue became separated from the main body of the army and enveloped. After some intense fighting where the English try to break through and the French try to break out of this envelopment, King John is convinced to surrender himself in exchange for safe conduct. From their he is taken to the English camp and presented to the Prince of Wales, who was the commander and highest ranking Englishman present, and then to England.

The military practice of allowing nobles to be ransomed was a fairly common practice at the time and was actually quite profitable. King John's ransom, for example, was negotiated to be in the area of 3 million livres total. This was several times the annual income of the Crown of France, but you could imagine that there would still be quite lucrative payments for other knights/barons/dukes, so soldiers and mercenaries had huge incentives to capture rather than kill these kinds of hostages on the field.

The second custom that is worth remembering is that killing a prisoner after they surrender falls under the war crime of perfidy, that is, acting in bad faith under the auspices of a truce. While I can't go into depth on the specifics of the more modern examples that you provide, in both cases, there was some kind of prosecution or criminal charge that was applied in order to punish the captive, and as I will explain later, it is not really in the English political interest to fabricate some reason to kill or mistreat King John. To do so would seem highly dishonorable by both the French and also by their own subjects. While there was no ICC to administer the rules of war at the time, the idea of honor in surrenders and truces is an old one and was valued because it protected your own subjects in the case that they were captured. So if the English mistreat King John after he has agreed to surrender on the battlefield and entered their custody, they risk a similar fate for any English nobles who fall into French hands. Now, just to expand a little more, it is worth noting that the customs around surrenders and chivalric honor were not absolute. Commoners and mercenaries, for example, were generally not granted this kind of treatment, so they had less reason to stop fighting or stick around a losing battle. Moreover, before an engagement, an army could indicate that they would not grant quarter to the losing side as a form of intimidation. This practice was most common during siege negotiations, but also occurred during battles, the most notable example being when the French Kings displayed their Oriflamme banner, which indicated that they would offer no mercy to their opponents.

Now, on to the political meat of this question. While capturing the French King is a masterstroke and extremely powerful, I would argue it is not something that ensures total victory. It is true that the theoretical root of medieval power and decision-making rested in the person of the King himself, but this does not mean that there are not other centers of power and it is not the same thing as controlling kingdom's infrastructure. Even after Poitiers, the Dauphin Charles (who was now the regent) and several of John's other sons were free and returning to Paris to assume control of the government. The bulk of the royal domain and many cities were still under the control of French garrisons, including Rheims, Paris, Orleans, Tours and Languedoc. Other territories were held by vassals of the French King, such as Flanders, Brittany, and Bourbon. In addition, there was another claimant to the French throne, Charles the Bad of Navarre who was free to continue to plot. So while Edward III could issue commands or force John to issue proclamations, there is no guarantee that any of these actors would abide by them. More importantly, the English were really in a position to seize any of these territories or control the other nobles since Edward's war up to this point had focused on chevaunchee rather than holding territory, meaning he really only controlled his duchy of Aquitaine.

What ended up happening is a complete disintegration of government, law and order. While the Dauphin was an extremely capable leader and politician in his own right, his was only the regent, and since his authority was not well-defined, he could not fully exert the powers of government like a king could. Mercenaries and local nobles roamed freely, peasants and cities revolted. Charles the Bad attempted to take and control territory for himself. All of this is extremely bad for France and King John, but it does not implicitly give control of the country to the English. What it does do is improve their negotiating position. What King John's challenge is to negotiate a treaty with Edward III that secures his freedom while also ensuring the all the aforementioned partisans play their part in that treaty.

So in short, from a political perspective Edward III doesn't control France per say, but having King John in his control prevents France from having any cohesive governmental action and thus improves his negotiating position with John and various factions in France. Unlike in other cases where the head of state of a country is captured, the English don't have control of the political or economic infrastructure of France after Poitiers. So what Edward III agrees to is a 3 million livres ransom, the transfer of several additional counties and cities to his control in full sovereignty (these were territories he had ancestral claims too, but traditionally the English had held them as vassals of France), and several hostages including another of John's sons to ensure the good faith of the agreement. In exchange, Edward III renounced claims to several other territories as well as to France itself, and King John was freed. In short, negotiating with King John gets Edward III tangible economic and political gains that he would not otherwise obtain, in this case his own garrisons and officials administering almost a third of France with no interference from the French King.

As for your last question about John's return to captivity, I am not aware of any definitive source or academic consensus on this question. The short background of this is that John's son Louis was one of the hostages turned over to the English as part of the treaty. Louis did not like his gilded cage and escaped back to France. John ended up doing the "honorable" thing and turned himself over to the English. While John was upholding the letter of chivalry and honor, there is really no arguing that this is a huge blunder from the perspective of realpolitik as John freely removes himself from the royal administration.

I hope this helps and let me know if you have any follow up questions!!