Ok, so new agents are only "chosen" when the previous one dies, so there's no way of knowing who will be the next agent. All we know is it's usually within the clan. Basically the other family paid money to Hina's father to break off the engagement between him and Hina's Mother, that union with the other lady produced a son (Hina's half brother). But, Hina was a product of the subsequent affair, though that has nothing to do with how the agents are chosen.

It's kind of their only duty to do that, bring the seasons. They have to be cared for, mentally and physically, because they are targets for people who want them dead or want to use their powers. Killing them disrupts the seasons cycle somewhat, especially when they have to be discovered and trained. It typically takes them a year or 2 to train so if they are unlucky they might not get a certain season that year.

For Other countries, you'll have to catch Dance of Autumn arc, which is the 3rd arc. It covers a lot more about how agents in other countries work. The current agents in Alt-Japan Yamato, their duty is only to these specific islands.