Some more updates on the server situation:
Host Europe replied that the only way to increase the socket limit would be if we switched to a bigger server package - which is not an option imo.
I've been looking at different options for limiting the number of sockets we need, but so far I didn't find anything that is both effective and simple.
Then there's the "apache bid farewell, cruel world" issue. Looking at older logs, this apparently happened regularly even on normal days, so one or two people every day have probably seen that ugly nginx 504 error page before now. The added load just caused it to come up more often and affect more requests. It would be neat to find out why it happens, but the workaround seems to be working, so you should only get a few seconds delay now where you would have got an error before.