A bigger alternator.
Defective alternator, dead cell in battery, or something is on pulling power from the battery.
To power your computer without a battery
Disconnect the battery cable, you do not have to touch the battery.
remove the battery and plug the power supply You mean running on AC power without a battery?
The answer will depend on the battery, and the amount of power it is required to deliver.
saves battery power without shutting down
Yes a 3.7v or 7.4v li-po battery can power a 5v device without spoiling it because P=IV.
No, the battery is part of it's circuit. In fact nothing in the car will work without the battery.
the function of a battry is to give somthing power without wires
No, as long as the power adapter is plugged in, the laptop will boot without the battery installed.
The iPhone battery was built-in, it is not so easy to reaplace it without open tool. But there are some extend power for iPhone, you can choose them to get more power.
You will need to use the power cord and/or the battery in order to turn the laptop on.