How can I check if Diablo III works on a friend's computer?
Solution 1:
You can do exactly as you suggested: re-download or copy your installation and install it (you don't need any code to do that), it's 100% legal.
You can then log on to your BattleNet account and try to play.
Solution 2:
The following website, Can You Run It, run checks on your computer (video card, cpu, memory, etc) and compares it against the minimum/suggested requirements of a ton of games, including Diablo 3. So far, it has provided me with good quality results.
http://www.systemrequirementslab.com/cyri/
What's most useful is the shaders check which tells you if the game will start or not.
Hope it helps.
Solution 3:
Anyone who bought the boxed version also gets 3 trial passes to give to friends. Could get one of them and give it to your friend if you don't want to give him your battle.net credentials. I can give you one if you would like. (Currently at work and don't have it on me but will be home @ 6 PM EST)