How long will items stick around when dropped

As Alissa said, the sign saying "take items as soon as possible" is simply there so you will remember to empty the box. Anything left in the box will just stay there when you return to the island, reducing the number of things you can take home again. It would actually be a way to take things with you to the island had they not disallowed putting things into the pick-up end of the box.

Don't worry, the game gives you plenty of warnings in cases where you might lose items: the island residents always warn you when doing something will sell off items implicitly, anyone sending you mail will remind you to empty your mailbox, because overflow mail is not (never?) received, and the game provides a lost and found (in the police station) for items removed from the ground because there was no room to drop it or because an event or new house is taking place.

Fruit doesn't actually spoil; what happens is the last time you harvest a Perfect Fruit tree, one of the fruits it drops will be rotten. Perfect fruit trees can be harvested a random(?) amount of times and the last time you shake it the leaves will all fall off and you should get a rotten fruit instead of one of the perfects. Ordinary fruit left on the ground can be left indefinitely without spoiling, best I'm aware.

Turnips, on the other hand, will spoil a few days after purchasing because of the "stalk market" depends on scarcity. Rotten turnips, while more expensive, may be easier to obtain if your only goal is the bugs that collect on rotten items.