I was able to look completely at your administrator page - you have a lot of 3rd part extensions installed , most probably there are some restrictions coming from one of them that avoid T3 Framework megamenu to work in the correct way - My suggestion is to disable one at once all of them and then find the one giving this issue,regarding updates i was able to verify they works correctly through Ja extension manager component or Joomla installer default -
Hope it helps.
Regards