In another posting you asked about setting up a proxy while the site was being migrated to new home. Did, by chance, you leave that proxy on and configured?
In a proxy setup there is a forward ... server accessed rec. a url and the proxy piece in between sorta re-points to the real service (URL) etc. There is also a reverse proxy configured so that the client that requested can rec. the page content requested via the URL client sent.
This last error screen sort indicates the proxy ain't working - errors can't be returned to client ... which would be a good protection thang running a proxy.
Is there a need for proxy still? Remove proxy settings that might be config.php file of Moodle. Removing those proxy lines in config.php doesn't require restart of services IF the proxy config of the apache server/service is OFF.
If you re-call ... you granted me access to your server some time ago to check things out ... I can login. I can click on admin menus and navigate around without issue of being logged off. My browser doesn't have any proxy settings in it and I don't get any errors like you are seeing.
Comment: ... been there done this ... my own worst enemy. By that I mean, doing things that aren't familiar or within current experience upon someone's (new provider) recommendation and it's not really needed. Then couldn't re-call how to un-do what I did! :\
'spirit of sharing', Ken