We can laugh about something, but to laugh at somebody or yourself is a fixed expression.
- Is "siebie" simply a more emphatic form of "się" as a reflexive pronoun?
- Is "z siebie" in a certain case or is it just part of an idiomatic phrase when using the verb "śmiać się?"
Seems that Wiktionary considers it to be a form of "się", but that seems a bit strange to me... well, frankly, it doesn't matter that much. Anyway, it would not be possible to use "się" itself here.
As "śmiać się" takes Genitive (śmiać się z kogo? z czego?), that's Genitive.
https://en.wiktionary.org/wiki/si%C4%99#Declension - declension of "się".
I would have thought the "z siebie" portion of the sentence to be redundant. Why is it necessary if się indicates reflexive? Couldn't I simply say "mężczyzna się śmieje"? What purpose does "się" otherwise serve?
"Mężczyzna się śmieje" would mean "the man laughs", but if he specifically laughs at himself, you have to add "z siebie" just like you'll add "at himself" in English.
That said, "się" alone doesn't work as "yourself". It can sometimes even just work as "one", for example: "Nie przechodzi się na czerwonym świetle" means "One doesn't walk [as in - cross the street] on a red light".
Check out this discussion: https://forum.duolingo.com/comment/16097555/Use-of-si%C4%99 where someone asks about the use of "się", it's been explained there in detail :)