Hi, welcome to the forum bama9411 - good questions, but probably not the answers you're looking for...
First, can points from special items such as the Cherry Blossom Tree be carried over to the next contest?
No, each leaderboard has stand alone items and their points are only for the current leaderboard.
For example, if I were to save the Cherry Blossom Trees I already have will I be able to chop them and still receive the extra 4 points they give or will they become regular trees after the contest is over?
They become regular trees. They won't give you points next time.
Also, will the special items that you can collect from every few hours or so still work for the next contest? Such as if I were to purchase the Serene Circle from The Spring Faire will I still receive points from it for the next contest?
No. same as point 1. the Serene Circle will revert to decoration at the end of the leaderboard. It won't be any good next leaderboard.