Will try and answer as best I can:
1) I don't know the abilities of your firewall, however, most apps can POST data to the web without the INTERNET permission. It would do this using something called an Intent
. But this would cause the browser to pop open. It would be pretty obvious if an app was doing this.
An easy way to be extra (extremely?) careful would be to install more than one browser and every call would ask you which browser to use (for all browser web requests, even legit ones).
But this is pretty unlikely to be a problem. However, having the popup about which browser to use will give you fine grained control (if that is what you seek).
2) Admob is not a spy network, it's Google's mobile advertising division. What you turned off is the tracking and targeting that Admob uses to customize ads to you. However, you will still get ads (unless blocked by your firewall). Google allows you to opt-out of tracking, not advertising.
hope that helps