Not able to access 3G Data when no Wi-Fi connection
that was weird.
the day after I posted this, I dialed the magic 4636 number, changed nothing in the settings and the mobile data was working allright. I was confused.

Then today, it happened again. 
Mobile data was unavailable, there was a ! over the signal strenght icon. I ran the Ping test and I got this:

Ping IPAddr: Fail: IP Addr not reachable
Ping hostname : Fail: Host unreachable
HTTP Client test: FAIL : IOException.
This also disabled the normal calling capability of the phone (see other post :error: call not sent. Unable to make outcalls?)

I turned the phone OFF and ON.
It booted allright and now I has H+ and full mobile data usage.

So I thank you for the help, and I hope the workaround will be good enough to close the issue for now.