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Either a discharged battery or a corroded battery terminal. The battery could be discharged because the alternator isn't working.
If it does freeze, no. The ice would damage the plates/circuitry maybe even the case.
If it were foolish enough to be out and about in winter then it definitely would. Northern snakes pass the winter in deep burrows.
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It depends on what type of battery it is. Ni-CD battery do need to be discharged completely before recharging it (I would recommend getting a discharger), Ni-MH batteries can be discharged but don't need to be, and Li-PO batteries should absolutely not be discharged completely, because they catch fire and explode when fully discharged (Ni-CD batteries have a memory effect, while Ni-MH and Li-PO batteries do not).
Maybe there is a short to earth somewhere in the car so the battery is being permanently discharged.
Discharged battery possibly or trouble with starter.
The leaves can not survive the cold. They would freeze. The trees become dormant and can survive the winter.
Because in the winter they would freeze, an in the summer it would be too hot.
They would freeze to death
I would guess if a battery has a water to acid ratio high enough to freeze that it would not hold a charge well. Be sure the case did not crack.
A car battery should not be "cycled" as in fully discharged and then recharged. Such a cycling would quickly destroy this kind of battery. A car battery is discharged slightly when you start your car, then it is recharged again when you drive your car. Proper charging whilst running car engine is good for the battery. A normal car-battery (Lead-acid) can last for many years if not "cycled". Never discharge battery more than 50%. How often you actually need to recharge your battery depends on how it is used.