The pro sports restrictions are because they have a union and a collective bargaining agreement. If it weren't for that all the draft, trade, free agency, and so on rules would be unenforceable.
Sort of. There's a nexus between antitrust and the unions, but in terms of why someone can't just quit and go work for another team, that relies mostly on what's called a "negative covenant" that's in all of the standard player contracts and related clauses and it involves jumping between leagues or even sports as well, so the anti-trust surface is a bit different or possibly not in the mix, and can also involve individual sports like tennis or boxing.
A lot of the other anti-competitive things sports leagues get away with are because of what you're talking about, but the reason every California-based professional athlete can't just rip up their contract and walk away is they have a different kind of reasoning going into it based on their rarity.