= Solution
Let $\ell(\theta;x)=\log p(x\mid\theta)$. The <maximum-likelihood estimator> maximizes this function; the <score function> is $U=\partial_\theta\ell$, the <Observed Fisher information> is $\widehat j=-\partial_\theta^2\ell(\widehat\theta;x)$, and the <Fisher information> is $I_n(\theta)=\mathbb E_\theta U^2=-\mathbb E_\theta\ell''$ under the stated differentiation regularity. Differentiating normalization gives $\mathbb EU=0$. For iid observations the score is a sum of iid centered terms with <variance> $I_1$, so the <central limit theorem> gives $U/\sqrt{nI_1}\Rightarrow N(0,1)$ when $0<I_1<\infty$. Under the additional usual consistency, identifiability and interior-MLE regularity, $\sqrt{I_n}(\widehat\theta-\theta)\Rightarrow N(0,1)$. Differentiation under the <integral> alone does not imply all these MLE conditions.
For the specified autoregression, index the innovations so $X_i=\theta X_{i-1}+E_i$, with $x_0=1$. Conditional normal densities give
$$
L(\theta;x)=(2\pi)^{-n/2}\exp\left[-\frac12\sum_{i=1}^n(x_i-\theta x_{i-1})^2\right].
$$
Set $A=\sum x_{i-1}^2\geq1$, $B=\sum x_{i-1}x_i$ and $C=\sum x_i^2$. Completing the square gives
$$
\boxed{\widehat\theta=B/A,\qquad\widehat j=A.}
$$
The likelihood ratio between two data vectors is independent of $\theta$ precisely when their coefficients $A,B$ agree, because its log is a constant plus $\theta\Delta B-\theta^2\Delta A/2$. The positive-density likelihood-ratio criterion therefore makes $(A,B)$ minimal sufficient. Since $(A,B)\leftrightarrow(\widehat j,\widehat\theta)$ is bijective, the requested pair is also minimal sufficient.
With the flat improper prior, the posterior kernel is $\exp[-A(\theta-\widehat\theta)^2/2]$, which is integrable because $A\geq1$. Thus
$$
\boxed{\theta\mid x\sim N(\widehat\theta,A^{-1}),\qquad S=\sqrt A=\sqrt{\widehat j}.}
$$
Conditionally on the observations, $S(\theta-\widehat\theta)$ is exactly standard normal, as is $E_1$. This is a posterior statement, not a claim that the estimator standardized by the random information is an exact normal sampling pivot.
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