Great question! In the big update I'm working on for the game I actually cut these mechanics completely for these reasons.
However your interpretation is correct for "Visit"! Each player gets an Easy, Medium and Hard challenge for their turn in the place they visit, their turn ends after these three challenges. The Assist mechanic is supposed to encourage players to stick together and help each other by making the dice rolls easier, so the "assisting" player gives the player who's challenge it is a bonus without losing out on their own challenges either.
I hope that clears it up for this version! The new version will maybe probably also be out before your online game as well, so keep an eye out for that.