The residual is the corresponding expression using the sample regression function. Assume that Xt is given and non-random. This means that it can be taken out of expectations. See the proofs of these properties in the appendix to Chapter 3. See the proofs in the appendix to Chapter 3.
Hypothesis testing requires that the error terms ut also have the normal distribution.
EX 3.5 a. X values being closer to their mean implies that Sxx is smaller. From Equations 3.18 and 3.19, we see that a smaller Sxx means a larger variance for the estimators.