= Solution
Let $x$ and $\epsilon$ denote the design and error vectors, and put $P=xx^T/S_{xx}$. The residual vector is $r=(I-P)\epsilon$, while $\widehat\beta-\beta=x^T\epsilon/S_{xx}$. Since $(I-P)x=0$,
$$
\operatorname{Cov}(r,\widehat\beta)=\frac{\sigma^2}{S_{xx}}(I-P)x=0.
$$
These are jointly Gaussian linear functions of $\epsilon$, so the entire residual vector is independent of $\widehat\beta$, not merely uncorrelated with it. Choose an <orthonormal basis> whose first vector is $x/\sqrt{S_{xx}}$. In that basis $\epsilon/\sigma$ has independent standard normal coordinates, and the residual keeps exactly the remaining $n-1$ coordinates. Therefore
$$
\boxed{\widehat\beta\ \perp\!\!\!\perp\ \widehat\sigma^2,\qquad
\frac{n\widehat\sigma^2}{\sigma^2}=\frac{\mathrm{RSS}}{\sigma^2}\sim\chi^2_{n-1}.}
$$
Together with part (ii), this specifies the joint distribution as the product of the slope's normal law and the scaled <chi-squared distribution>. In particular $E\widehat\sigma^2=(n-1)\sigma^2/n$: the variance maximum-likelihood estimator is biased downward.
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