Check the transmission neutral switch or the connection to it. The neutral switch is on the lower left side, beside the kickdown rod. There is also a connector in the wiring harness on the right side of the engine, at the rear of the right side valve cover. If the neutral switch isn't functioning for any reason, the symptoms will be exactly what you described. I once did some engine work on my 88 Maxi and forgot to connect the neutral switch in the harness, so I speak from sad experience...;-)
Yes
That light doesn't mean that your battery needs to be replaced. It means that your battery is not being charged while the car is running. Have your alternator tested.
you may have a bad battery or a bad voltage regulator
An alternator is designed to keep a fully charged battery at peak level. It is not designed to charge a dead battery. Installing a new alternator on a car with a dead battery can in some cases ruin the new alternator. Fully charge the dead battery before starting the engine.
As the alternator runs, it charges the battery.
As long as the battery is fully charged it will start no matter the condition of the alternator. Eventually the battery will run down if the alternator is not keeping it charged.
If you replaced the battery with a brand new one, then no. The battery comes from the store with a fuul charge. If it didn't come charged up, then you have bought a defective battery.
A battery is nothing more than a storage device for electric power. The alternator changes the kinetic energy given off by the engine into electrical energy. If the battery can be charged and store energy but the battery keeps discharging after you use the vehicle, check the alternator. If someone accidentally reversed polarity EVEN BRIEFLY while jump-starting the vehicle the diode plate is probably destroyed and you will have to replace or repair the alternator.
An alternator has diodes which transform ac voltage to dc voltage so that the battery gets negative and positive current on its poles thus making the battery to be charged.
As long as the battery is charged
It is there to keep the battery fully charged.
Yes it can as long as the battery is charged . The purpose of the alternator is to put the power used back into the battery.