the computer will shut off entirely
No
get a new laptop
it does from experience....
Laptop batteries generally have a lifespan of 3-4 years depending on usage.
Not unless it is plugged into a wall outlet.
It base on the model of your laptop, laptop battery has it's own model too. And different battery's model may fit one laptop model. Your question answered by battery specialist hklika.
The battery is dead.
Either a dead battery - or a faulty power supply.
Your battery will lose charge faster.
The laptop will turn off when it is unplugged from the AC adapter
not really. if you take out the battery when plugged in you could short circuit the laptop. trust me i know.
Yes, you can. Most laptop will work fine without its battery, some of them will not. Dell is usually of those which do not have such problems.