MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

Question

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$.

1. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased?
2. Does there exist an unbiased estimator of $P\{X_{i}=1\}$ whose variance achieves the Cramer-Rao lower bound?

My thoughts:

1. MLE of $\lambda$ is $\bar{X}$. Thus, MLE of $P\{X_{i}=1\}=\lambda e^{-\lambda}$ would be $\bar{X} e^{- \bar{X} }$. Because $E\bar{X} e^{- \bar{X} }=\lambda e^{-1/n-n\lambda+n\lambda e^{-1/n}}\ne \lambda e^{-\lambda }$, (Made a mistake before, as the expectation calculated as $\lambda e^{n-1-n\lambda+n\lambda/e}$) the MLE is biased.
2. I have calculated the Cramer-Rao lower bound as $\lambda(1-\lambda)^2 e^{-2\lambda}/n$. But I don't know how to start from here. And how to show whether there exists an unbiased estimator generally?