No, but it is generally required to patch the game on each PC unless you install a pre-patched archive from http://anywhere.impulsedriven.com. The game does not have online activation off the disk, it's a voluntary CD key registration to your Impulse/SD account to allow you to get the latest updates and re-download the game whenever you like from Impulse/Impulse Anywhere.

Yes, however as you (probably) don't own all of the machines on the LAN, SD asks that you uninstall the game after the game. There is no cd-key-in-use check on LAN games, only Online multiplayer.

You don't have to re-download. You can download on your PC and make an archive (right click in My Games > Archive Application). Then you can restore the archive with Impulse on the other PCs, saving a lot of download time. Archives created in Impulse however need an internet connection to restore (Impulse needs to be logged in for you to be able to access the restore function). Alternatively, you can download the fully-patched archive from the Anywhere link above, which has its own installer and doesn't require an internet connection to unpack.

If you make an archive through Impulse, you can change where it's saved and all that in the Impulse prefs through the top left menu dropdown.

You need to have registered your key before you can download them from Anywhere. It only shows games/apps that are registered to your account. I've never installed off it, but I don't think you need your key after it's registered to your account.