Just bought it (kinda late, I know) and I'm loving it.

Let's say I want to install onto another computer. Do I still need the cd key?
Also, is it possible to copy the game directly straight to another hard drive and have it playable?

If I am not mistaken the Enhanced Edition is DRM Free - You may want to check the version and add the director's cut patched version.

The retail version (aka DVD, not GOG) installs version 1.0 with Tages. The game then updates itself, which removes the Tages integration. It updates up to 1.4 or 1.5, I don't remember. The NA version also requires the "director's cut" patch to get it fully uncensored, as well. There's no internet connection requirement or anything. And it ends up DRM-free by the time you're done installing it, but it is technically DRM'd for a couple of minutes during the install process.

I don't remember whether it uses a CD-Key or not, but provided you still have the disk you can install it onto a second computer, yes.

You might need an internet connection when installing so the game can update itself and remove the DRM. If you have absolutely no internet connection, you'll have to get the patches manually ahead of time and then run them as soon as you're done installing. There are instructions for where to find the patches elsewhere in this forum.

EDIT: But yes, once the game is updated and the DRM is removed, you can absolutely play without an internet connection. And you can install on as many machines as you like.

The retail version of the Enhanced Edition has all the patches up to 1.4 on the disk. You'll need an internet connection for patch 1.5 and the director's cut patch, but you don't need it on the installing computer; you can download the patches on another computer and put them on a USB drive or something. So if you have a computer that never connects to the internet ever, it's still possible to get a fully updated game.