I know that in the list of rewards it doesn't mention any kind of reward, such as an engram or ascendant materials, but do you really earn absolutely nothing at the end of the nightfall if you've already done it that week? Nothing randomly drops from the last boss of the strike?
No, you are rewarded only once. Nightfall only gives rewards at the end of strike. Before you go first time, you can see that it has 'Unknows Rewards' in Director, but has none after completion.
You can still get random engram drops from enemies during the strike, but that's not nightfall specific. Strike bosses do not drop gear in Destiny, but you can get race-specific item from them (e.g. Ether Seeds or House Banner from Sepiks Prime, etc).
So the answer is: yes, you gain absolutely nothing at the end of nightfall strike if you've already done it.
The quick answer is No you don't get anything for running the Nightfall Strike more than once before the reset. (It resets every Tuesday.)
One thing I'd like to add though, is that you can run the Nightfall Strike 3 times a week if you have 3 different characters to run it with. I have 1 of each class and I run it once with each of them each week before the reset.