Disks should be stored vertically in a box. Leaving them horizontally one on another is not a good solution. Some disks had plastic transparent foil sleeves in which they could be stored for longer time, but they aren't popular now. Before inserting disk to the drive sleeve should be removed,  if not, drive will be jammed.

If you try to use higher density floppies in lower density drives (like PC disks in Amiga), you may have problem: Data written in higher density drive will be usually readable, but erasing them and writing data in 720K drive may cause errors.
Why? Higher density drive has stronger head, writing bits stronger. Weaker head can't remagnetize higher magnetized surface.
There's only one solution: To use 1.44MB drive formatted disks only to put data TO low-density drive system, but write on this system to completely magnetically blank disks (after formatting in low-density drive), these disks should be never written with a higher density disk drive.
Making these blank disks is easy: Just demagnetize the disk entirely. To do it, you can use AC electromagnetic coil or even transformer soldering iron. Standing 2-3m away of media power electromagnet or soldering iron on. Slowly move it near the disk, make few swinging/round moves few centimeters near media. Repeat it for about 30 seconds, may be longer, but DON'T TURN IT OFF. Before turning off move coil/iron 2-3m away from media, or it'll stay magnetized some way.

In general, older disks of good manufacturers may be better and may have less errors than newest, as manufacturers decided to push floppies away from market as fast as possible by manufacturing poor media.

To read a PC disk to a computer for archiving purposes, you should use an old PC with DOS (Norton Commander's "Copy Disk->To Image" or RawRead), Windows 9x (WinImage or RawRead) or even Linux (dd utility, think before you type!), but avoid using WinXP and later systems. These systems have very poor floppy disk drive handling routines, probably to make users not use floppy disks and buy USB flash drives. This will give you many problems, false read errors and other things like this. USB floppy drives are even worse - they are slower and won't read old disks reliably. Forget about 720K or XDF, even DMF format with USB drives - most of these won't read it, some may even damage disks.
Apple computers may use USB drives (tested in Mac OS X 10.4 and my PowerPC G4 MDD), but not for lower-density disks - These disks may become irreversibly damaged instantly after inserting to USB floppy drive, even with write-protection tab set! Of course floppy routines in Mac OS X 10.4 are even slower than in WinXP. Many times you'll get few "tries" to read a disk every second, this is completely normal thing. Unmounting a disk 2 times is quite normal.