Club A pays Club B a fee for ownership of a player. Club A then has to negotiate a contract with that player for the deal to happen.
There are two times a year where players can be transferred between clubs as well. The summer window runs June to August and the winter window runs the month of January.
Is any player open to be transferred? Is the fee negotiable? Can club B match the offer, or is it the player's choice?
Yep the fee negotiated between the clubs. The fee paid to the club not to player. Some players will get a "signing" bonus but it is not nearly as common as it in American sports. A piece of image rights are big in the contracts.
In a sense, but the sports landscape is set up much differently than in American sports. There is a set heirarchy that really hasn't been changed for decades. The same huge clubs are the big buyers and everyone else feeds into that. Occasionally you will see shifts, both negative (Leeds' financial implosion) or positive (Chelsea, PSG and Manchester City getting bought by extremely wealthy owners) but for the most part, the same twenty or so clubs have been the best for the last four or five decades. Baseball is the closest, with teams (usually small market) out of the hunt for the playoffs and trying to sell their current assets for ones that may produce more in the future. In soccer, it's generally smaller clubs selling to bigger clubs. They then take the money they get from the bigger club and invest in new, cheaper players or to help pay down debts. On occasion you'll see a big club sell to another big club but that usually only happens when the player wants out of his current club. You'll also see big clubs sell to smaller clubs when they buy players that don't work out at a reduced price from how the bought him.
All leagues have "trades". Player swaps are not that common the other leagues as they are in the MLS.
Yea you really won't see MLS teams just straight up paying each other for a player. They use allocation, DP spots, and players as leverage. Not sure I've ever seen any MLS team trade with a non-MLS team though.
There's also different rules in Euro leagues in regards to the amount of foreigners or non-EU passport holding players allowed on a team or the minimum they can be paid. This has always been a negative for American players. England for example has no foreign limits, but the UK has strict work permit laws that says any non-EU player has to have played in 75% of their national team's "A" games over the past 1-2 years. Although sometimes they'll be granted a Work Permit if they're close to that. There's also the "special talent" exception that pretty much every South American gets for having one name. In the Netherlands, there's no limit on non-EU players but they have to be paid a minimum of 500K euros a year. French clubs are only allowed 4 non-EU players iirc