Multivariate distribution of linear regression coefficients and unbiased variance estimator Excerpt from "Elements of Statistical Learning", p.47
Assume that the conditional expectation of $Y$ is linear in $X_1, \ldots, X_p$. Also assume that the deviations of $Y$ around its expectation are additive and Gaussian. Hence $$Y = E(Y \mid X_1, \ldots, X_p) + \varepsilon,$$ $$ = \beta_0 + \sum_{j = 1}^p X_j \beta_j + \varepsilon,$$ where the error $\varepsilon \sim N(0,\sigma^2)$ is a Gaussian random variable.
It is then easy to show that $\hat \beta \sim N(\beta, (X^t X)^{-1} \sigma^2)$ (1) and that $(N - p - 1)\hat \sigma^2 \sim \sigma^2 \chi^2_{N-p-1}$ (2).
Earlier they also assume that the observations $y_i$ are uncorrelated and have constant variance $\sigma^2$, and the $x_i$ are fixed.

Question
This part of ESL is repetition of basics that I'm trying to rehash since it was a long time ago that I studied this stuff.
For (1) I can show that $\mathrm{Cov}(\hat \beta) = (X^t X)^{-1} \sigma^2$ and of course $E(\hat \beta) = \beta$, but don't I need to know that $Y$ is normally distributed to be able to tell the distribution of $\hat \beta \sim N(E(\hat \beta), \mathrm{Cov}(\hat \beta))$?
For (2) I know that the hat matrix $X(X^t X)^{-1}X^t$ is idempotent, has rank $p + 1$ and $X^t \varepsilon = 0$. How can I finish (2)?

The normality of $\hat{\beta}$ follows from the fact that any affine transformation of a normal random vector is still normal.  Here, in matrix form, $\hat{\beta} = (X'X)^{-1}X'Y = \beta + (X'X)^{-1}X'\epsilon$ is an affine transformation of the $N$-dimensional random vector $\epsilon \sim N(0, \sigma^2 I_{(N)})$. Hence the first result.
The second one regards the distribution of the residual vector
$$\hat{\epsilon} = (I_{(N)} - H)Y = (I_{(N)} - H)(X\beta + \epsilon) = (I_{(N)} - H)\epsilon,$$
where $H = X(X'X)^{-1}X'$.
Since $(N - p - 1)\hat{\sigma}^2 = \hat{\epsilon}'\hat{\epsilon} = \epsilon'(I_{(N)} - H)\epsilon$, $\sigma^{-1}\epsilon \sim N(0, I_{(N)})$, the second result follows from the distribution of quadratic form of multivariate normal random vectors (see, for example, Theorem 1.4.2 in Aspects of Multivariate Statistical Theory by R. Muirhead).

For your convenience (and also because of the importance of this theorem), I copy it here:

If $X$ is $N_m(\mu, I_m)$ and $B$ is an $m \times m$ symmetric matrix then $X'BX$ has a noncentral $\chi^2$ distribution if and only if $B$ is idempotent, in which case the degrees of freedom and the noncentrality parameter are respectively $k = \operatorname{rank}(B) = \operatorname{tr}(B)$ and $\delta = \mu'B\mu$.

The idea of the proof is that $B$ has the canonical form $B = H\operatorname{diag}(I_{(k)}, 0)H'$ with $H$ orthogonal when $B$ is idempotent and rank $k$. The closedness of multivariate normal distribution under the affine transformation again plays a role here.
