Yes, IF the battery is not damaged [both internally as well as externally], AND it is recharged properly.
Dead battery
Dead cell in battery or something is on pulling power from the battery. Disconnect the negative battery cable overnight with the battery fully charged (12.6 volts). If the battery is dead the next morning it has a dead cell and needs replacing. If however the battery is still fully charged then a light is on somewhere or a relay is stuck closed.
If by bounce you mean like a rubber ball bounces: The bounce, or not is determined by the mechanical properties of the battery, and these doesn't change with the battery being charged or not. Dead, or fully charged, a battery will behave just the same when dropped.
the more the battery is charged, the less water. dead battery is close to 100% water. charged battery is close to 90 - 100% sufuric acid
Something is on draining the battery or it has a dead cell and needs replacing.
A Compaq nc6000 will run on ac power when the battery is dead. The battery is long lasting but it can be charged by using ac power.
As long as you keep them charged you do not have to remove them. A fully charged battery will not freeze, but a dead battery will.
New batteries almost always come fully charged. If it was not then charge it with a battery charger and do not use your vehicle alternator to charge a dead battery. You may do harm to the alternator which is not built to charge a dead battery but is instead built to keep a good battery fully charged.
An auto motive can be recharged by connecting the dead battery to another vehicle with a charged battery by the use of a jumper cable by connecting both end to both batteries then starting the car with the charged battery inside.
Dead cell in battery, loose of corroded battery connections, or the charging system on the mower engine is defective.
Yes, once you have jump started the car (from another car's battery), driving the car will recharge the dead battery (provided the battery is OK). This is a terrible idea and may damage the alternator. Drive the car immediately home and charge the battery with a battery charger. Alternators are designed to keep a charged battery fully charged. They are not designed to charge a dead battery. This puts way too much strain on the alternator circuits.
The hazard lights get their power from the battery so if the battery is dead they will not work.