Let $X_i$ be a random variable.
Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$.
Let $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^{n}X_i$.
Let $\hat{\sigma}_n^2\geq 0$ be a consistent estimator of $\sigma^2$ as $n \rightarrow \infty$.
By CLT, $\frac{\bar{X}_n-\mathbb{E}(\bar{X}_n)}{\sqrt{Var(\bar{X}_n)}} =\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma} \rightarrow_d N(0,1)$ as $n \rightarrow \infty$.
Using $\hat{\sigma}_n^2 \rightarrow_p \sigma^2$ as $n\rightarrow \infty$ and applying the Slutsky's Theorem, we can conclude that \begin{equation} \sqrt{n}\frac{\bar{X}_n-\mu}{\hat{\sigma}_n} \rightarrow_d N(0,1) \end{equation} as $n \rightarrow \infty$.
Let $\{\epsilon_n\}_{n=1}^{\infty}$ be a sequence of positive constants approaching zero as $n \rightarrow \infty$. Hence $\tilde{\sigma}_n^2:=\hat{\sigma}^2_n+\epsilon_n$ is a consistent estimator of $\sigma^2$ as $n \rightarrow \infty$.
Therefore, \begin{equation} \sqrt{n}\frac{\bar{X}_n-\mu}{\tilde{\sigma}_n} \rightarrow_d N(0,1) \end{equation} as $n \rightarrow \infty$.
Questions:
1) Are all steps above right (in particular the final convergence in distribution result)?
2) How fast the sequence $\{\epsilon_n\}_{n=1}^{\infty}$ approaches zero is relevant somewhere? If it is relevant, how can I set that rate of convergence?
For 1) Yes, they are both correct.
2) no, the rate is irrelevant since its a sequence of constants that disappear in the limit.
Is there a counterexample that gave you concern?