Why is it Schlagzeuger and not Zeugschläger? Does it really mean someone who begets beats (rather than someone who hits stuff)?
No, that would be Schlag
erzeuger, which means someone who begets electric shocks. Schlagzeug means percussion, so a Schlagzeuger is a percussionist. You don't say percusser, either. Zeugschläger is good, though.
What happened to the middle of selbständig? (Especially considering that they've now put back the midddle of Nussschale, Rollladen and Stillleben, for example.)
Some people do write selbstständig, but since selb- is indeed a sufficient portion of selber and selbst, nothing is actually missing.
Why stattdessen and währenddessen and deswegen but trotzdem?
Because different prepositions require different cases: statt, während, and wegen take the genitive, while trotz takes the dative. Wait, that isn't even true. Hm. But it must be at least a part of the explanation.