Point and Interval Estimation

Suppose we have a random sample ( $X_1,X_2, ... ,X_n$) drawn from a Normal random variable $X \sim N(\mu , \sigma^2)$.

Why are the following estimators of $\mu$ unsatisfactory?

i) $X_1$

Do I just need to prove biasedness for this? i.e. $E(X_1) = X_1 \not= \mu$?

Also I have another question ... is $\bar{x} = \mu$ i.e. $E(\bar{x}^2) = E(\mu^2) = \frac{\sigma^2}{n} + \mu^2$ ?

Thanks any help would be appreciated.