= Solution
Set $\mathbf e=\mathbf1/\sqrt n$ and $\mathbf f=\mathbf x/\sqrt{S_{xx}}$. These are orthonormal because $\sum_i x_i=0$. Complete them to an <orthonormal basis> $\mathbf e,\mathbf f,\mathbf q_3,\ldots,\mathbf q_n$ of $\mathbb R^n$. An <orthogonal transformation> of the standard normal error vector $\boldsymbol\varepsilon/\sigma$ has independent <normal random variables> as coordinates $Z_1,\ldots,Z_n$: its <joint probability density> remains $(2\pi)^{-n/2}\exp(-\|z\|^2/2)$ because the transformation preserves squared length and has <Jacobian determinant> of absolute value one.
Projection onto these <basis> directions gives
$$
\widehat\alpha=\alpha+\frac\sigma{\sqrt n}Z_1,\qquad\widehat\beta=\beta+\frac\sigma{\sqrt{S_{xx}}}Z_2,\qquad\frac{\operatorname{RSS}}{\sigma^2}=\sum_{j=3}^nZ_j^2.
$$
Therefore the <joint distribution of least-squares and variance estimators> is
$$
\boxed{\widehat\alpha\sim N(\alpha,\sigma^2/n),\quad\widehat\beta\sim N(\beta,\sigma^2/S_{xx}),\quad\frac{n\widehat\sigma^2}{\sigma^2}\sim\chi^2_{n-2},\quad\text{all three independent}.}
$$
The last law is the <chi-squared distribution> of a sum of $n-2$ squared independent <normal random variables> with mean zero and variance one, and <independence> follows because the three statistics use disjoint independent coordinate sets. When all $x_i=0$, omit the slope direction: $\widehat\alpha$ and $n\widehat\sigma^2/\sigma^2\sim\chi^2_{n-1}$ are independent, and there is no identifiable slope estimator.
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