So you got a lot of questions here, and I'll try to answer them all. I'll focus on hockey, which shares traits with the other three (MLB, NBA, NFL) sports in North America, while also being unique.

As I understand it, most sports teams in Europe were founded as clubs by various groups, and they grew into the modern teams that exist today. That of course is a very simple understanding of how things went, and I'll be happy to get into it more if requested.

For North America, it was initially very similar in hockey: teams were representatives of various clubs and groups of people, grouped together either by profession, ethnic origin, or simply a desire to play sports. However this model changed in the early 1900s, as distinct franchises established specifically for hockey were developed.

The reasons for these vary, but it mainly comes down to money: unlike many other sports, which can be played on a field or something easily accessible, hockey required ice. Initially arenas used natural ice, in that they simply flooded the rink and let nature freeze it; this meant that hockey seasons were short, lasting roughly form January to March each year. Developments in artificial ice allowed the expansion of this, but due to the costs associated it was not until the 1920s that all NHL arenas were using it.

With the high costs of maintaining an arena, not every club could afford to keep paying the fees associated, and thus rich backers were required to front teams. That it could be a profitable venture also helped, and so by about 1910 most top-tier hockey teams were owned by a group, and not just a club. The ever-increasing costs have ensured this method has lasted, and NHL (and nearly all hockey teams) have been privately owned since then.

As to youth teams, there actually was a sort of youth system in hockey, for a while. The earliest teams did have junior clubs, and would call up talented players to join the senior team. However the debate over professionalism in Canadian sports meant that this would not last long-term: by 1908 top-level hockey was openly professional, though players had been covertly paid for years. The organizing bodies for junior teams maintained their amateurism, and with strict codes guiding that (even playing a professional in a game could cause an amateur to lose their status) meant it was much more rigid.

However starting in the 1930s and picking up in the 1940s and 1950s, NHL teams began to sponsor junior teams: they would pay the costs for the team, and in return had exclusive rights to the players once they became professional. This system lasted until 1963, and it was slowly phased out by 1969, when junior teams were no longer sponsored and all junior-aged players were no longer bound to a pro team. This is what led to the modern draft, which was originally called the "amateur draft" but since 1979 has been known as the "entry draft".

The draft exists to give all teams an equal chance at junior players. While teams had sponsored junior teams before, it was heavily slanted towards a couple teams that had extensive networks, and thus were able to get all the best players. Other teams were not so fortunate, and thus spent years as the worst teams in the league. To remedy this the draft was created, and it was formatted that the worst team was able to pick eligible players first, followed by other teams in reverse order of the standings that year (this system changed in 1993 due to allegations of "tanking", meaning teams deliberately losing in order to pick higher. It's since been further modified and a lottery is held for the first picks, but the worst teams have the best odds). Any eligible player, North American or foreign, could be selected, though for the past 20 years or so players not selected are declared free agents and able to sign with anyone (there are more restrictions than that, but that is a short version of it).

Lastly, a player selected in the draft does not have to play for that team, and indeed there have been cases where that happened (Eric Lindros is by far the most famous example; he was selected first overall, but refused to play for the team, and was traded a year later for a massive return). However there are regulations to enforce that, and while they have changed over the years, currently teams hold the rights to a drafted player for 2 years, at which point he is free to re-enter the draft (they can re-enter once, and then are free agents).

I think that covers everything here, and if there are points anyone would like expanded on I'll be glad to do so.

So major league baseball probably is the closest thing to the European football system, but to your point it's still very different. While there isn't any relegation / advancement between leagues, there is a major league, and 7 levels of "minor leagues" that players have to advance through (although they don't have to play in all of them, and teams could just straight promote kids to the majors if they wanted) to get to the majors. Teams can sign kids as young as 16, so to an extent you can think of it like a lesser version of the youth academy system over in Europe. I'll go into that progression and some of the various nuances shortly, but I think it's important to contextualize things first.

One of the major differences I've seen between European football and North America sports in general is the notion and desire for parity. While obviously you have the Yankees, Dodgers, Red Sox, etc, you don't have teams utterly dominate things the way a Real Madrid/ManU/PSG/Milan etc. While even non baseball fans know who the Yankees are, they haven't won a world series since 2009, and that represents their only championship since 2000. Why is this? Well, leagues work hard to try and make sure teams have a level of equality of opportunity. They do this by having a salary cap, the amateur player draft, the international free agency system, revenue sharing, and a few other things I'm probably neglecting to mention. So while there's definitely a disparity in resources, prestige, whatever you want to call it, you can have more teams rise and fall, and in the past 10 years, 7 different teams have won the World Series.

The first mechanism to achieve that is the salary cap. Lot's of North American leagues have some sort of salary cap in place to make sure that the rich teams can't just go out and build a team of galacticos and run over the small market teams. In baseball it's actually called the luxury tax. Teams are actually able to spend as much as they want, but if they exceed the luxury tax threshold, there's a series of progressive penalties. For the 2020 season teams the threshold is set at $208 million. A club exceeding the threshold for the first time must pay 20% on all overages (ie, if they spent 218M, they would have a 2M tax hit). If a team exceeds it for a 2nd consecutive year, they have to pay 30%, and for a 3rd consecutive year 50%. A key point here is that the penalties work consecutively, so if a team dips below the threshold the clock resets. There's also steeper penalties in place for teams who go really over the tax, as those percentages jump up, and they could loose picks in the amateur player draft. Now there's a lot of debate in the baseball community now about why teams aren't just blowing past the threshold and eating the penalties, and there's lots of theories out there about why that is, but since they implemented this system, most teams have tried to stay under the tax.

Beyond this luxury tax you have another feature that would look a bit strange to a European onlooker. There's 2 main ways for aspiring baseball players to make it to the big leagues. First is the amateur player draft. Every year the league holds an amateur player draft for high school and college (university) students from the US and Canada. There are 40 rounds (millions of american kids/young adults play baseball and teams can have anywhere from 8-12 teams to fill with players) and the first 10 have an "assigned value" which serve as a guideline for how much their bonus can be. In an effort to again ensure parity, MLB sets a bonus pool of how much total money teams can spend on draft picks based off how they did the previous year. Teams who do worse get to draft earlier, and have larger bonus pools with which to sign their picks. Teams face penalties for going over their bonus pool, so generally they try to stay under. Actual signing bonuses are heavily weighted towards the higher tiers, with the top draft pick getting 5-7M (for reference the top NFL draft pick this year got 36M) and that dropping down to 125K by the 10th round (and much lower for rounds after that).

Players who get drafted don't have to sign with the team that drafts them, but then they have to wait for the next years draft to come around to get redrafted. It's common for players drafted outside of the first round (especially high schoolers) to not sign, play a season in college and then come back and try to improve their draft status. One of the major factors teams look at is "signability" for their top tier picks so its not terribly common for 1st round picks not to sign.Now you'll note when I started talking about the draft I noted that it only applies to American and Canadian players. For everyone else, there is an even more complex and convoluted system known as the international free agency system. For players born outside of the US and Canada, they can be signed by teams as soon as they turn 16. There is a defined bonus pool of money teams can spend (it starts the same for every team, but can change based off a few different factors and teams can also trade bonus pool money as part of other transactions) and teams face stiff penalties for going over or negotiating with kids who are younger then 16. While this is probably the closest to the academy system, there are a few more rules and protections in place. There's definitely an inequality dynamic with multi billion dollar franchises negotiating with teenagers from some of the poorest parts of Latin America. The top prospects can get a few million dollars for signing, but the super majority of players are signing for under 10K.

Due to the popularity of baseball in Japan and Korea, they have some different rules with how players can make the jump to MLB if they so desire. There's not a ton of them but you can read about the process here if you so desire.

Once a player signs with a team, either via the amateur draft, or the international free agency system the team has a few years of "control" over them. While under team control, players salaries are basically set, and they are forced to play for that team. There is a salary arbitration period towards the end of team control where players can earn more based off their performance; but they can't just say "I don't want to play for the Yankees anymore" and test free agency. MLB League minimum for 2020 is $563,500 which gets paid out during the course of the regular season. During the minor leagues players make anywhere from $10-90K a year depending on the level and how much time they have in the system. Minor league guys are also only paid during the season and generally live in pretty crummy living situations during the season as $10,000 isn't really enough to live off of in America, especially with a lot of Latin players sending money back to their families.

Because of the huge disparity in level of play across the amateurs (highschool and college), and due to them using different equipment (hitters use metal bats in high school and college; and wood bats in the pros) it takes players a long time to reach the majors. It takes players years to hone the skills necessary to even make it to a major league stadium. You've got both a domestic and international rookie league, short season (starts right after the mlb draft for kids that get drafted to get their feet wet), Low A, High A, Double A, and finally Triple A. Of course with each level the competition gets stiffer and stiffer. Even your top picks in the draft generally take 3-5 years to make it to the big leagues, if they ever make it. For international players it can take even longer, as lot's of them aren't even physically mature by the time they're signed.

Once players have accrued six full years of playing time at the major league level, they're eligible for free agency, and can sign those mega $330M deals you probably read about in the news. The last 3 years of team control, players are eligible for arbitration, where they can basically say "hey I've been really good and I should make $X" and teams can either agree to their proposal, or go to arbitration. It depends a lot on what position they play and how good they are, but once they hit arbitration they'll go from making league minimum to a few million or more (the record for 1st year arbitration is 11.5M, and 27M for 3rd year) for that year. Teams can sign players before they hit arbitration, and often do, but when you have a maximum amount of money you can spend, it behooves them to find value on players producing a lot while getting paid a little.Players on the other hand, are constricted by balancing money in their pockets with belief in their own potential future earnings. As a 21 year old from a poor family, do you take an 8 year contact worth a guaranteed $100M? Or do you wait 3 years, hope you don't get hurt or loose your ability, and try and get a deal worth 20-30M a year? Up until that year 6, players don't have a ton of options if they're unhappy with their situation. They can try to demand a trade, but teams don't have to pay them more, or oblige their request.

Hope that answers your question as it pertains to baseball. Happy to delve more into a particular area, or answer any follow ups!