What is the null hypothesis checked by ANOVA? I know that null hypothesis used by ANOVA is that means in all the groups are the same and, if p-value is small, we reject the null hypothesis which means that we believe that not all the means are the same (that is, there are groups with different means).
However, it is not clear to me if we also assume within the null hypothesis that standard deviation in all the groups is the same. What about higher momenta (like skewness and kurtosis), do we assume them to be equal? Do we assume, within the null hypothesis, that distributions of all the groups are the same? Do we assume that the distributions are normal?

I think, "assume within the null hypothesis" is not a valid statement. We do not assume nothing in null hypothesis. We check if hull hypothesis can be rejected. And to be able to check it, we make some assumptions. These assumptions help us to develop formulas for test statistic and p-value.
So:

if we also assume that standard deviation in all the groups is the same

Yes, we assume this. It means that all the formulas used in testing procedure are valid only if standard deviation in all the groups is the same. If it is not true we should use some other test (Welch ANOVA probably) that do not assumes it.

What about higher momenta (like skewness and kurtosis), do we assume
them to be equal?

No.

Do we assume that the distributions are normal?

Yes. Just like with SDs. It means that all the formulas used in testing procedure are valid only if distributions are normal. If it is not true we should use some other test (Kruskal-Wallis ANOVA probably) that do not assumes it.

Do we assume that distributions of all
the groups are the same?

If we assumed it, we wouldn't need any test. Because, if all distributions are the same, all means are the same too.
In case you meant "Do we assume that distributions of all the groups are from the same family (like all are normals or all are exponentials)?", look at previous paragraph.
