Apache issues resolved, sites back online
Yes, finally, with a little help from the cPanel team, I got Apache/PHP to rebuild correctly, and my sites are again working properly!
Of course, I am still working on the slowness issue. Even with page caching, Apache and suPHP thrash the CPU for several seconds the first time a large dynamic page is loaded. I'm trying to narrow this down to either server configuration or a VPS issue.
I have definitely experienced this before on an over-allocated VMware array, so it's a possibility my web server is just on a busy host box. But I'm still tuning things on the server and trying to minimize the load in every way I can, because more often than not these things are caused by configuration oversight or lack of proper tuning (and user naivety perhaps, in my case).