Hi to all,
Well after 4 years my problem with connecting to arcgis.com has been solved.
Trouble was in our firewall and after IT adapted the "white list" all was fine. It took all of 4 days to fix.
I had given them the list 4 years ago and was told that "this can't be It". I had ESRI support send me another copy of the list and finally got some cooperation.