Regarding the first issue: there was another site on this server that was causing high I/O load (the harddrive could not keep up). It was probably at around the time of your post that I was trying to move the site to another server. Unfortunately as the site in question is a free image hosting site, the backups were a few dozen gigabytes in size, so during the move it taxed the harddrive even more.
As for the second issue, I upgraded APC (Alternative PHP Cache, an application that can speed up page requests) to the latest version, which appears to be badly broken causing "Unable to open more files" errors which to visitors manifested themselves as "No input file specified" errors. This problem happened rather unexpectedly, so I didn't notice it for hours.
I apologize for these problems, but both should be solved right now.