Obviously, I underestimated the severity of the situation earlier. Just to be sure, the new server is now a model which should be able to handle further growth in the foreseeable future.

The site has been down for a few hours again. This time, it was due to high CPU load. I'm trying to figure out the problem, but of course, it is possible that such a complex set of scripts combined with increasing number of visitors could just be too much for our current hosting. I'll keep you updated.