Yes, you're right.
You show that it's unique by construction. The MLE is found by computing the log-likelihood and finding its stationary points (where the derivative with respect to lambda is 0), you also have to show that the second derivative is negative, therefore showing that it corresponds to a maximum of the likelihood. In this case there's only one stationary point, and so this is the unique MLE.
How did you find the MLE?
As for the last question, a MLE is unbiased and I know that it is more asymptotically efficient (has a smaller mean squared error) than any other unbiased estimator, but I don't know how to prove it.