The initial cause of the fault could be a simple battery fault requiring a new battery, or it could have been caused by an alternator fault.
Get the alternator output tested at ideal and at higher rpm under load (cheap and quick, easy DIY job if you have a cheap and basic automotive multimeter). To do it correctly, you'll need to fit a new battery to begin with. If the output is faulty then replace the alternator immediately to avoid ruining your new battery. Note: NEVER test the alternator by removing a battery connection: that will definitely ruin the alternator rectifier pack.
To answer your initial question, once the battery is effectively totally dead, your car will run until the alternator rectifier pack blows.
if it starts, it can drive until the gas runs out. the battery is only there for 3 reasons. it is there to store the power needed to crank the engine over, to supply higher amperage when needed and then to absorb the ac voltage that has bled past the diodes in the alternator. a dead cell is a sign that the battery is on its last leg.
Not very long.
The cell is usually shorted inside, so in time it will drain the remaining 5 good cells. This usually happens when you most need it (starting on a cold morning, driving with everything on...).
Very idiotic question. if a cell was dead, the battery wouldn't work.
Forever. The alternator would power the car once it was running.
Until you get tired of having to get a jump start every time you go somewhere and replace it.
