Unable to access internet
100% that it matters if your phone is T-Mobile and your provider is Suncom. The phone has a TMob. firmware installed in it, so it is configured to TMob. infrastructure....in other words, TMob does not a Web Session. It might have something similar(parameters) that MIGHT work on Suncom, but that's a guessing game...
As for the Unlocking part, it just means that you can put a Suncom, or any other SIM card and it would work...(mainly for voice calls). Try to find a phone Flashing shop and have it flahed with Suncom's firmware.
Best of luck....
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