If I miss a day of running the challenge (today is Molten Scar), is it still available the next day, alongside the new challenge, or at reset, does it become unavailable when Stone Forest becomes available?
Nope not available the next day
They go based on a days rotation. Only Sunday's have all 3 open. Same for the Mountain. Sunday is your best bet!
So the short of it, is that there is no advantage to skipping a day.
With a contest starting tomorrow that's partially based on gathering scraps, thought if a challenge would linger until the next day, there would be a strategic reason to not farm it. Guess not!
Although, technically, the contest starts two hours before reset, so if I wanted to stay up incredibly late, or wake up incredibly early, to hit the challenge up, I could do that. But I'm not that dedicated to this game.
Have some kids. Then you'll be up 2 hours before reset whether you want to or not....