Does ‘Yo’ in ‘yo mama joke’ actually refer to “Your” mother? Why couldn't it simply be (stupid / fat / ugly) "mama joke"?

Solution 1:

"Yo mama" does mean "your mother." The "yo mama" jokes are more or less humorous insults, used as a sort of verbal sparring on the street. Without going into its origins, the way this arose amongst those who created it was this: a person's mother is sacrosanct, an untouchable icon, so insulting someone's mother was a particularly effective way of getting the better of that person. What you are seeing is a watered-down version for the purpose of making it more generally accessible and more humorous, so the reference to someone's actual mother is diminished, but not to the point of turning it into generically "any woman."

And given the nature of the material, I for one can't see it being extended into other versions such as the ones you suggest.

Solution 2:

Yes, 'yo momma' is a non-rhotic pronunciation (dropped-r) of 'your momma'. Note the usual exposition is

yo momma so fat/stupid/...

which also has dropped the copula (the 'is'). In standard English it would read 'your mother is so fat/stupid'). Though I don't know the history of the joke, this sounds like AAVE (African-American Vernacular English) or also Southern American English from the southern US. AAVE and SAE have many similar features (the above two plus monophthongization, like sounding 'Ah' for 'I').

'yo momma' jokes are always referring specifically to your mother. On e might tell the joke as 'She so fat...' or 'That man so stupid...' and then it is about those people.

As to extending, no, the pattern is simply 'yo momma so X that ...' and that's it.