I was able to fix the issue regarding the crash every time I emptied the cache. I rebuilt Apache/PHP with easyapache, then built matching PHP w/ LS.
That seemed to have fixed that portion of the issue.
However, the remaining issue now appears to be this:
2012-04-02 12:47:25.618 [INFO] [126.96.36.199:58306-0#APVH_redacted.com]
connection to [/tmp/lshttpd/lsphp5.sock.615] on request #500, confirmed, 0, associated process: 29101, running: 1, error: Connection reset by peer!
This appears every 5 minutes in the error log. Exactly the same error (on request #500)
I've disabled APC and XCache as well.