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That fall and the shock from it most likely broke a run or a supporting member and/or a grounding connection. That qualifies for 'it broke', given the symptoms.
It will need servicing to work again, or at least opening and a superficial examination as to what looks broken-off inside, and whether any circuit board cracked (not easy to repair, unless it's just a corner and a couple of connection runs, or less... in some circumstances.)
Perhaps. It depends upon what's broken inside your TiVo. Opening it and examining it for cracks, breakage, etc., would not void the warranty - there is no warranty on damage from accidental dropping or falling, in general.