Make sure the battery cables are making good contact with the battery and are clean and tight.
A battery charges nothing it accepts a charge from the alternator.
Check the fuses
its the baterry
batteries are stored energy, nothing should have happened. They say to disconnect the negative terminal for safety reasons.
Something is causing a drain on the battery even with the vehicle off and nothing turned on. A good mechanic can do a drain test on the battery to see what may be draining the charge. A dead cell in the battery itself will also cause the battery to go dead. Disconnect the negative battery cable overnight and if the battery is dead the next morning it has a dead cell and must be replaced. If not look for a light that is on or a relay that is stuck.
It is the alternator. I have a 95 Monte Carlo and this has happened to me twice in the 3 years I've had the car. Each time I had to have the alternator replaced. The first time I had a problem with the alternator I took it to a shop and had it fixed. Two years later the alternator went out again without warning. That time we took it to a different shop and bought a new alternator with a three year warranty so it'll be free if it goes out two years later again.
Yes, the battery starts the engine, and as long as the battery is fully charged the engine will start. The battery will loose it's charge if the alternator is defective and not replenishing the charge of the battery but the alternator has nothing to do with starting the engine.
Yes, because a bad coil will not allow the engine to run. But if the engine is running then the coil has nothing to do with the alternator charging the battery.
no, the light has nothing to do with the alternator working properly, the light is only to warn you that it isn't charging the battery properly.
No, the battery is part of it's circuit. In fact nothing in the car will work without the battery.
The alternator may need to be replaced.
Could be a bad belt, if you bought a rebuilt alternator then it may not be up to snuff, you might need to replace your battery, or you might have a parasitic drain in the vehicle's electronics somewhere which has nothing to do with the alternator. Without knowing the conditions under which it occurs, or why you had the alternator replaced in the first place (namely, whether you actually diagnosed it or just threw a part at it), it's a bit impossible to narrow it down.