how to fix grey wifi icon (no internet), and how to track what caused that?
Grey icons don't mean that you don't have internet connectivity. eldarerathis's answer explains:

It has to do with whether or not you've currently got a good connection to Google's servers for sync services and the like. From page 27 of their Android 2.3 Users Guide:


  Network status icons turn green if you have a Google Account added to
  your phone and the phone is connected to Google services, for syncing
  your Gmail, Calendar events, contacts, for backing up your settings,
  and so on. If you don’t have a Google Account or if, for example,
  you’re connected to a Wi-Fi network that is not connected to the
  Internet, the network icons are white. I'm not really sure why it
  bounces back and forth between green and white sometimes. I've noticed
  it on my phone but everything works fine (I run CyanogenMod, though,
  so perhaps it has to do with that). It could also be that green means
  it's currently in the act of syncing, but the wording in the document
  is a little vague, in my opinion.


In the case of Honeycomb or Ice Cream Sandwich, the colors are blue/grey instead of green/white, but they still have the same meaning.