I cannot move any file to /home/
Exactly. You shouldn't be able to with the way OSX ships.
I recall recently somebody mentioned that it is not a good idea to store things in /home/ as it is supposed to be used by the OS to store system files. That is why I got an error when I tried to move things to that directory.
As a user, you should only be storing things in your home directory (i.e. /Users/<your username> and not in /home) unless you really know what you're doing. It's certainly possible to use /home but the way OSX ships by default, the operating system "owns" /home for directory services (according to /etc/auto_master and /etc/auto_home). It sounds like you don't have a reason to use /home on OSX so I won't go into more detail.
What if I am also the root (administrator) user? Is ~/ the same as / ?
No. On OSX, the root user's home directory is /var/root. Other operating systems might have it someplace else, like at /root. Most Mac users never interact with this directory because OSX ships with the root user disabled (and it should only be enabled if you have a really good reason to enable it and you really know what you're doing).
BTW, if you really want to learn more about how these sorts of things work, I recommending doing it in a virtual machine and not on your main OSX install. It's easy to mess things up once you start playing around in Terminal. Download VirtualBox and install Linux or FreeBSD and go nuts. If you mess things up, it's no big deal because you can re-install. Even though OSX is a different operating system, once you understand how one Unix (like) operating system works, you'll have a much better understanding of how all of them work.