1.mp4 players with nokia type batteries are prone to do that. try changing the battery. if you feel that the battery is moving too much, try inserting small pieces of folded paper to make it fit right.
if it all fails, than its a hardware problem and you should contact the seller to replace it/refund.
2. to know if your battery is full or not, just have a look at your player when it is charging and if the bar in the battery is not going up and down/across, then it means it is full.
3. no you cant mod firmwares above 9.0.48 yet.
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