Hypothesis testing: understanding test statistics and their sampling distribution I am struggling a bit with the real meaning of the test statistic and their sampling distribution, but I cannot find my answer. So, I am looking for clarifications. Sorry in advance if it seems very basic.
To illustrate my question and to keep it simple, I will take the example of a one-sample t-test: $t= \frac{\overline{x} - \mu_0}{se}$
When we say that test statistics have their sampling distributions, does it mean that (in this case), the sampling distribution of the t-test is drawn from sampling distribution of the mean $\overline{X}$? Or is it drawn from the distribution of the null hypothesis assuming the mean of our sample $\overline{x}$ is fixed?

The meanings that you propose in your question are not wrong but problematic, because you make imprecise and somewhat confused use of terminology. You may get more insight carefully distinguishing the different meanings of the word "mean", see below.
The null hypothesis model is $X_1,\ldots,X_n$ i.i.d. with $X_i\sim N(\mu_0,\sigma^2)$. A very important distinction in statistics is between a true, but unobserved parameter value of the underlying distribution we are interested in, and an observable test statistic that we use to estimate it or do inference about it. In English, it seems that mostly, somewhat confusingly, the term "mean" is used for both the true underlying unobserved expected value $\mu_0$ and for the observed data mean $\bar x$, which is a fixed value and modelled as realisation of the random variable $\bar X$. Note that there are three kinds of means now, $\mu_0, \bar x, \bar X$, the latter two distinguished by whether we have already made the observations and computed the mean (that's $\bar x$, a fixed number but only fixed after having observed the data), or we're talking about the general method to compute the mean from our observations whatever they are (that's $\bar X$, a random variable).
The sample distribution of the t-statistic is derived from the null hypothesis distribution as you state, however the "fixed mean" of it is the underlying true $\mu_0$, not the $\bar x$.
Assuming this, there is a sampling distribution of the random variable $\bar X$ as computed from the $X_1,\ldots,X_n$, referred to as "sampling distribution of the mean", but note this is $\bar X$, not $\bar x$, because the latter one is a specific mean value of a specific sample.
The sampling distribution of the t-statistic $\frac{\bar X-\mu_0}{S/\sqrt{n}}$ (note my use of capital letters here; we can only speak of the sampling distribution of random variables, which, if you want, stand for methods of computing things rather than already computed fixed values) is strongly mathematically related to the sampling distribution of $\bar X$, but some more math is required to bring in the connection to the estimated standard deviation $S$, which is another random variable (if required, one can also use lower case $s$ for the actual value it takes after having collected the sample, and $\sigma$ for the unobserved underlying true value of the sd). The wording "drawn from the sampling distribution of the mean" isn't really appropriate here; the idea is that random variables $X_1,\ldots,X_n$ are drawn from the original null distribution, and $\bar X$ and $S$ are computed from these random variables, which implies that their sampling distribution has the stated form, such as $t_{n-1}$ for the test statistic, and $N(\mu_0,\frac{\sigma^2}{n})$ for the mean random variable $\bar X$.
I know I am somewhat redundant here, but I hope it helps to review the key difference between true parameters, random variables, and values computed from samples at each step.
