GoDaddy was of little help - they did FINALLY confirm that the live site load time issues were due to "latency on specific nodes on the grid that they are working to smooth out" but blame the 500 server errors on Wordpress.
This doesn't make sense since the server errors appear at the same time that the load times jump and only when I am performing some action on the admin side (deactivating a plugin, activating a theme, etc). Denied any issues (filtering, firewall, etc) that could impact PHP activity through port 80. Denied any database issues.
I do think I found the SSH info while exploring in my hosting panel, so going to attempt to fix the other issue of the mystery PNG file now while I wait to see if I get another server error, having reactivated a group of plugins.
I really appreciate everyone's input and suggestions thus far. My hope is that this is a GoDaddy issue that they refuse to acknowledge and that the grid "latency" issues were and are the source of the server errors when I was trying to work in the backend.
