There is a 3.01 Firmware update for the A7R5. No features - Improves the operational stability of the camera.

I can't really help you with that but unfortunately, the internal batteries do degrade over a period of time.

Sounds to me like you have a dodgy contact on the camera lens mount. As you have already cleaned them etc. the only thing I can think of trying before you send it away for repair is to get something that's not too hard, such as a toothpick, and just try depressing slightly each contact to see if any are stuck or something.