It looks like you have several errors in your SP toolbox, some dating back before 5.5.11 (assuming that’s what you meant by ‘the last upgrade’), but the ones post update are all ‘error: Lost connection to MySQL server during query’. I don’t know too much about why you would loose connection to MySQL, but one possibility is exhausting PHP memory which is something you should be able to increase yourself.
It is also possible I guess that the database could have been corrupted somehow or the same thing happened while updating something and things weren’t properly updated / installed. We would need a second opinion on these errors.
Have any changes been made to any Simple:Press templates / functions? Are you running a child theme or standard?
To say ‘in the Admin backend admins can no longer edit members avatars’ has me a bit confused. delete or add to the admin pool? I’m not getting any errors on the panel page but didn’t want to delete or change anything.
Are you talking about editing a users profile on the forum page itself? If so I am seeing a jquery conflict on the profile page (amongst others) which clearly references something to do with the TinyMCE editor.
If you’re running any plugins that add to or change the TinyMCE editor in any way even for writing blog posts they could be loading their own jquery library instead of using the one supplied with WP which will almost certainly cause problems. Please disable anything that might fit the bill and see if things work.
Not sure what you mean by admin support pages either, have checked and admin panels seem OK with no errors and popups all work just fine. Could you be more specific?