Dead cell in the battery or something is on pulling power from the battery.
You left the light or something on last night and ran out the battery or your battery died
No, that will have no effect on the battery.
Yes, and disconnecting the battery while the engine is running can severely damage the computers
plates in the battery are arcing together. causing battery to short out and get hot. could explode. or you have a dead short in system. both will cause battery to get very hot.
the engine is running checky after starting
Typically, a bad or "going bad" alternator.
Possible the battery is just too old, or it is damaged.
It can, if you leave the lights or flashers on too long.
the alternator is out
If the cause of the dead battery is from the charging system failure, yes.
The Optispark distributor is notorious for causing this problem.
Not likely, but running the engine with a dead battery can damage some alternators. Always fully charge the battery when installing an alternator.