I didn't solve the mystery, but I had to delete the plugins folders via cPanel and add them one by one. I couldn't install the "missing" ones while all the folders existed.
BTW, while WP was telling me that there were no plugins, they were still functioning and doing their work.
I still haven't found out why every php file on the server now has a dump of Zend code at the beginning. My cursory attempts to read up on it tell me that it's the best thing since sliced bread for programming but not why it needs to attach itself to every PHP file.