Estimation Theory - Basic Question
The expectation of $x[n]$ is

$$E(x[n])=E(A+w[n])=E(A)+E(w[n])=A+0=A\tag{1}$$

because $A$ is deterministic and the noise $w[n]$ is assumed to have a mean of zero. So in that formula you sum up $N$ terms that are equal to $A$, and then you divide that sum by $N$, leaving you with the value $A$.

As for the expectation of the estimated value $\hat{A}$, note that $\hat{A}$ is a random variable because it necessarily depends on the data $x[n]$, which are random, so computing its expected value (and its variance) makes sense.