Go into options, device, advanced system settings, host routing table. What do you see? Is this blank or full of listings? If it's blank click menu (BB button) and register now. A message should arrive and then the host routing table will fill up with the relevant entries.
If you get the message check your signal again. It should say 3G with the Blackberry logo next to the 3G. That would then mean it's working.
The fact the G in your 3g is small would indicate that you have a regular data plan and not a Blackberry plan, or that it's not set up correctly at the least. Also the fact you're adding contacts on BBM but there are not actually adding indicates the same.
If the above doesn't solve the problem contact your network/carrier and ask them to activate a Blackberry plan on your account.
It should come back straight away so it looks to be that you don't have a Blackberry plan and with a generic data plan and not a Blackberry plan I'm afraid lots won't work including the problems you are describing above.
Contact your network/carrier and ask them to check which data plan you have. If, as I suspect, it's ust a generic plan then you'll need to swap it to a Blackberry plan in order to get the best from your phone