OK, I've figured out one piece of the puzzle. I have two SIM cards. One had credit for data access, the other didn't. The one that didn't - a TravelSIM - needed firstly a credit topup and secondly the APN "send.ee" needed to be set up specifically for the TravelSIM.
Once that was set up, the greyed out area for ...
>Mobile network> Enable data acess over mobile network: ON
... was no longer permanently grey, and I could turn it on or off.
Internet browsing and syncing on the phone is then no problem.
So maybe that was your problem, pottsie?
Mind you, I still have the problem that even though the phone says "connected", the PC it is connected to says "No internet access", and browsing via the PC doesn't work!