Pretty sure the days don't affect anything.

The difficulty is set by the biome you're currently in, and the monsters that spawn there.

There are game events (monster raids) that are triggered randomly, and those are set by the bosses you've beaten. Some events start only after you beat a certain boss, and continue only up to the point where you defeat the next boss, then they stop. Other raids become a permanent part of the game after that, and can randomly happen at any time.