This concept doesn't really exist in pro/rel football (soccer) leagues. With very few exceptions the system of main/farm team doesn't exist. Clubs have youth teams at different age tiers (with U23s and U18s being the most prominent). By and large these youth teams take part in their own, separate, youth-exclusive league/tournament/structure. (Edit: Although U21 players are usually always available for first-team selection without taking up a registration slot.)
The most notable exception is the Spanish top tier, La Liga. There, the top clubs' equivalent of an U23 team is named [Team Name] B (e.g., Barcelona B, Real Madrid B, etc.), and these teams play in the same pro/rel league structure as their top level club. They are not permitted to play in the same division. If Barcelona B would qualify by position to be promoted to La Liga, they would be passed over and it would go on to the next team. To my knowledge it is somewhat rare for a B team to even make it into the Segunda División; the third division (confusingly, the Segunda División B) has 4 such "B" teams.
I don't know the specifics of how the Spanish leagues operate, but generally speaking (like in the NHL) there is a limit on how many players can be registered to the team in one season, and windows during which registrations can be added or removed. These requirements are typically waived for players under a certain age (this age is usually 21, regardless of playing history); so it is conceivable that a good youth (U21) player could play regularly for both the main and B team in a given season. Other players assigned to the B team, however, who do not meet this eligibility would have to spend the entire season at their B club.
Most other major European leagues don't incorporate youth/B teams into their league structure though. The major objection is the desire to avoid diluting the competitive structure (every B team from a top tier club in a lower division means there's one fewer club, probably 100+ years old, that can't be in that division). If young players can't break into their parent club directly, they are often sent on loan to lower league teams for a season at a time (although 2 seasons is becoming more common) [European teams often have to do this with players requiring work permits as well; it is faster/easier to gain residency in certain countries than others]. These can rarely involve formal parent/"feeder" club arrangements, wherein the parent club pays the lower league club for the relationship and the feeder club accepts players on loan (with a cap on how many you can have total and from one club) from the parent club without being responsible for their wages. Usually loans are to clubs with no formal relationship, though. That club is under no obligation to play the player. Players can only be called back to the parent club if such an option was agreed upon. There would be no moving back and forth (there is a limit on how many teams you can be registered for during one season). Some top tier clubs have gotten to the point where they have literal dozens of young prospects out on loan at any given time (not to the same team).
And yep, as @NerveDamage says, transfer of player ownership (really transfers of the player's registration) almost exclusively take the form of money changing hands. Player-for-player swaps are almost nonexistent and there is no such thing as draft picks. When a transfer takes place, a new contract must be negotiated between the player and the purchasing club. This means that selling a player (or renewing) becomes critical as their contract nears its end (like the NHL) but there is no need to consider if a player 'has term' as the purchasing club; they'll be on whatever length of contract you agree with them (usually 5 years).
Correct. Regional conferences only exist at low levels of football because the teams are too poor to travel far.
It's even worse in Argentina. Relegation is tracked on a rolling three-year (I think) average so that the good teams (who are actually solvent) don't accidentally get relegated because of one bad season.
Disparities between tiers on a pro/rel pyramid can be really drastic. English Premier League teams who get relegated to the Championship (the 2nd tier; which is confusingly ahead of the "First Division" [tier 3] and "Second Division" [tier 4]) actually receive what are called 'parachute payments' to ease the financial contraction of going down a division (where their wages are now way too high to sustain with the decreased revenue).
Pro/rel is a great system in a sport that has been using it for decades (and has been playing the sport for over a hundred years). But the US has too many clubs for a single, unified top division and too few to properly split it into tiers, not to mention absolutely no cultural or historical concept to understand it by. It's probably (slowly) coming to US Soccer, but that's a whole other can of worms.
BTW, Detroit City FC is great fun and y'all should go see their games!