Endless Summer mode is Day 0 on the event board, and should unlock when you've collected 12 badges!
As for the difficulty, yes, it does ramp up steadily as you move down the event board. With some of the larger levels, especially, there is a chance you will have a few campers you might not have a chance to rescue before they are caught - the game does try to minimize this possibility, but because of the RNG it will still come up occasionally.