Two well-known "fill-the-pattern" traditions are the children songs
1) Meine Oma fährt im Hühnerstall Motorrad where there is something like a "standard" set of stanzas, but enthused singers may spontaneously create new ones describing various other strange activities of the grandmother. (A recent variation I met was Meine Oma hat im Rachenraum Coroooona, Corooona...)
2) The Scheiße song with stanzas like Scheiße auf dem Autoreifen macht beim Bremsen braune Streifen, and endless variations what effects Scheiße might have when put in other places. (Scheiße auf dem Weiberbusen hält die Männer ab vom Schmusen would be another well-known one. A third one is Scheiße auf dem Autodach wird bei 180 flach. You see the pattern.)
These songs are popular with eight to twelve years olds, approximately.
3) You correctly mentioned the ... außer [name]... verses such as Alle Kinder sind im Bett. Außer Rainer, den mag keiner.
For obvious reasons the better of these verses are rarely put down in writing, and the really spontaneously created anyway aren't.
4) There is some pattern gag featuring a bunny with a speech defect uttering sentences by the meaning "Do you have..." (Haddu...). These are known as Häschenwitze, but I cannot comment on them because I have never understood the joke. There are, however, many examples published on websites. Just use Häschenwitze as a search term. From my perspective, these are popular with 8 year olds.
5) A commenter above mentioned Gstanzl, a form of traditional music in the Alpine region where singers (often people gathered in the local pub) take turns in spontaneously adding stanzas, typically ironically commenting semi-public events relevant for the auditory (traditionally from life in the village, but nowadays it might shift to general politics). For the original meaning of the name Gstanzl, take as a hint stanza.
6) There is a tradition of Was ist der Unterschied zwischen... jokes (What's the difference of...) which then are to be answered with - ideally - rather unexpected solutions. Example: Was ist der Unterschied zwischen Hunden und Zwiebeln? - Es gibt keinen. Die Hunde bell'n und die Zwiebeln. (I admit that this is a particularly absurd one. But it is documented since at least 1977. I think I read it in the satirical monthly Pardon, and I somehow associate it with either Robert Gernhard or Friedrich Karl Waechter.)