Washing machine drain suddenly backflows
Alright, it took a bit of effort but we managed to take apart the drain construction (without breaking anything of course) so we could get deeper to unclog. Everything from the hose entry up to the neighbour's drain was already checked. The clog seems to have been a bit farther ahead, right about where my "diagram" stops. Lots and lots of gooey black gunk came out, I think we emptied about 3 or maybe 4 full buckets worth of the stuff.

We ran a couple quick tests with the washing machine: "rinse" program followed immediately by "centrifuge and pump" as soon as it stopped drawing water. This went perfectly fine for about 3 times (the vent pipe stayed dry the entire time), so we decided to pour a little drain unclogger down the top to get rid of any residual gunk. Then we ran the 2-program cycle twice again to rinse it out. My normal laundry is being washed as I'm typing this, so thanks for the tips and additional information on how drains work. =]

tl;dr: babby's first plumbing job was a success.