I started typing up a huge post about partitioning and migrating OSes, it started to shape up into a whole essay so i gave up, sorry
. Though i will go on a limb and say that there's been a lot of stuff written about everything you're asking about by better writers than me so it never hurts to go out there and read some documentation
Withough further ado, lets get into it.
I'll answer the simple portions though:
/mnt is a generic directory where pointers for your device mounts are stored. To access a storage device in linux you gotta mount it onto something. For example a CDROM without a disk inside is useless but if you want to read data off your CDR you burnt few weeks ago you put it into the drive and mount it onto /mnt/cdrom folder. The names are arbitrary and are there simply as a convention and convenience. You can mount the same CDR anywhere else you want as the system does allow you to do that, /tmp, /home would all work. But /mnt is generally considered the 'proper' place to store the mount points though. Plus you are guaranteed not to break anything by mounting (as that overlays the contents of the folder onto which its mounted) as no applications ever put files into that directory tree.
Second part of mount is the /etc/fstab file (which i come to as answer to your last question). Fstab is a file that describes all the information about mounting your filesystem. How, where, what device it is, etc. So for example if you were to try to access a cdr from the last post. You could do something like 'mount /mnt/cdrom' and the system would go read in /etc/fstab file, find out the cdrom entry and know that it needs to mount the device /dev/cdrom1 read-only onto the directory of /mnt/cdrom with autodetected filesystem. By default only root user can mount or unmount devices. but there's an options column in the fstab file for every device and if you want your users to be able to mount and unmount specific devices you can add 'users' as one of the options there.