Using of "at the turn of" with words other than "century" or "year" (or similar)

can I say something like, for instance, "at the turn of the beginning of the school year"

No. “The turn of the ” implies an end and a beginning (in that order) so to repeat “of the beginning” is a tautology.

For the same reason, you cannot say “at the turn of the beginning of the XX century"

You can say “at the turn of the school year"

or "at the turn of his arrest", to indicate a period immediately before and after his arrest?

No. “At the turn of XYY” is approximately “about the time that XYY was changing to XYZ” and you will see that the arrest does not change.

PS

I have a doubt regarding the use of "at the turn of".

This is not idiomatic. You should use “I have doubts (plural) regarding the use of "at the turn of".” Doubts is always plural in this set phrase – it doesn’t matter how many you have.