Is it possible to test my windows phone 8 app on an actual device, like Nokia Lumia 820 (NOT Simulator) before submitting to store? How?
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You can test your app on any real device, you just need to have developer account to unlock your phone or phones for debugging.
It costs 99$ per year per developer, or you can get it for free if you are student using DreamSpark, or BizSpark, or you have MSDN or TechNet subscription.
How to register your phone for development
Once you have developer account, just start the Windows Phone Developer Registration tool, attach your phone to PC, unlock it and then you can select the Device in your project for debugging.