Little things piling up that may have a common cause

This is a test of your diagnostic powers, Joomapolis. In a savagely ironic twist, if I enable max reporting, the site template breaks, so I am reduced to describing symptoms.

1. Group menu button next to groups' name does not do anything other than "click." This occurs on both my superadmin name and the other you have creds for.

2. This is puzzling: the user status field values, but not the title, disappear if a group is created by the user. I did this a couple of times to be sure.

3. Integrations are also getting mysterious as well. Komento displays their commenter profile correctly using the CB avatar but not on the comments themselves.

Any idea?

No physic powers here ... please provide Joomla/CB exact version numbers.

When you say Group button are you referring to GroupJive?

Rats on the psychic powers.

Joomla = 3.4.1 but issue was also present in 3.4.0
CB latest nightly build 2.0.7

I did mean GJ menu button, yes; on profile tab

In a savagely ironic twist, if I enable max reporting, the site template breaks

This is something you need to fix first. You most likely have some errors that are crippling your site causing a chain reaction of problems.

Cascading problem sheets, eh?

Ignoring the lost .css, the only error I see is this

Strict Standards: Declaration of getLatestEvents::getDisplayTab() should be compatible with cbTabHandler::getDisplayTab($tab, $user, $ui) in .../public_html/components/com_comprofiler/plugin/user/plug_cbjevents/jevents.php on line 489

Disable non-core plugins 1 by 1 within CB > Plugin Management to see if your issue goes away. This will help see if it's a plugin causing it.