I'd probably broaden this beyond 1/2 - I think the base case is the host gives you a chance to gamble with a question or test of skill, and the result is purely dependent on the player. The swap-box scenario is then an extreme case of that where the result depends less and less on the skill of the player, eventually reaching 50% chance of winning. I wouldn't say evolutionary-optimised, but maybe familiarity with the game-show tropes being somewhere along this scale. Monty Hall is then a twist on this extreme case, which pattern-matches to the more common 50% case with no allowance for the effect of the host's knowledge.

I do not think this is correct. First, the host should only offer you the chance to switch if you are winning, so the chance should be 0. Second, this example seems too contrived to be something that we would have evolved a good instinct about.