I went on to find out about the shared caching problem, and there does indeed seem to be a problem. I have usually 1 to 3 php-cgi processes running, so if APC is not shared, this could explain why sometimes my cache is emptied (interestingly, when that happens, total requests in the apcinfo.php page also starts back from 0).
So, I set up nginx with php-fpm just to test (I want to keep apache, but want to leave it intact at the moment), and there indeed the APC cache hits is close to 97% (instead of about 80%)
I was surprised though that some performance tests showed that apache with mpm-prefork (as is currently the case) did 500 php requests in about 45s. With nginx, the same 500 requests took 61s. I'm unsure why this could be, and I first need to work out why that is before I could consider switching from php in cgi/fastgi to fpm/fastcgi. Do you have any experience with this?