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.
http://acwiki3ds.wikispaces.com/Post+Office
I had to check it myself after I got the piggy bank at 1,000,000 Bells instead of the 10,000,000 it was in ACWW.
I can confirm that you receive 0.5% interest on the first of each month (I received 20,000 Bells on 8/1/13 for 4,000,000 in the bank).
Unfortunately I only have 7,000,000 in the bank so I cannot confirm anything past the aluminum case reward (which is indeed at 5,000,000 Bells). What strikes my interest is that you only have to save up 100,000,000 Bells to get all rewards, so it is quite feasible to simply save up that much money at the beginning of the game before doing anything else.
Update: Confirmed all rewards (ADB @ 100,000,000 Bells)
Best Answer
You can't take anything to the island except what you're wearing. No tools or items of any sort except clothing items that you are currently wearing. Note that wetsuits you are currently wearing count as clothes (so you can take them) but can still be used as tools on the island. Though this isn't a huge advantage as you can rent a wetsuit on the island anyway.
Tools instead will be provided by a Gyroid at the island and on each of the tours. You can take any non-tour related item back to your town by leaving it in the basket in the hut when you first reach the island. All other items will be sold at an extremely low price when you go home or go on a tour.