2G has internet access. As long as you see the 1X indicator. However, it will be miserably slow (its pretty close to dial-up speed-wise).
When your phone's asleep, its basically in a low power mode. The processor goes slower in order to save battery. All your cell phone service connectivity and such is still active and things are still running on your phone. So yes, you will still get updates and such, just as long as you haven't set it to turn off the internet while it sleeps.
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