You earn medals by participating in Tours on Tortimer's Island. You can either do this solo, with friends, or through random matchmaking through the Internet.
The number of medals you are awarded is dependent on how well you do in each minigame. The better your performance, the more medals are rewarded. There are also 3 difficulties, from 1-3 stars, and the higher the difficulty, the more medals you earn.
For example, during the Easy Hammer Tour (1 star), I hit the machine 91 times, which gave me a Gold Tortimer award (these are bronze/silver/gold and aren't tangible, just an award to judge how well you did) and 6 medals. During the Level-Up Hammer Tour (2 stars), I hit the machine 80 times, which gave me a Gold Tortimer award and 8 medals.
In addition, when playing with multiple people, you earn Bonus Medals based on how well you did in comparison to them. The way the games work is that they are a mix of cooperative and competitive. You all have to do something that works towards a collective goal, but you do get bonuses based on how well you did. So, again using the Hammer Tour as an example, you have to hit the machine 30 times within 3 minutes. Each person's hit would go towards the total, but at the end, they would also get bonus medals for how many times they hit it.
Once you earn 50 medals, you can earn membership to "Club Tortimer", which allows you to participate in Tours with random people on the Internet. If you're not a member, you can only go on Tours with people with whom you are friends. This is handy if you don't have friends that you can play with or they're on at times when you're not available, etc.
In conclusion, the fastest way to earn medals is by playing with multiple people as opposed to solo, and playing on higher rather than lower difficulties (and performing well, to boot).
Based on past animal crossing games and judging from a week's worth of experience, items will persist forever when dropped until picked up, and they will be moved to the "lost and found" (does this exist in AC:NL? I haven't seen one yet) as soon as an event or new house needs to occupy the space the item took. In the original AC I left items all over several acres of land and they were only removed when (oops) an event cleared the acre.
I haven't noticed any dropped items disappearing so far. Note that buried fossils and gyroids are an exception to this rule; naturally placed buried items will be removed the next day.
Best Answer
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.