Yes it will
It is one of hundreds of things that could turn the check engine light on.
A low battery can cause the electrical system to flash check engine lights
yes if one of the wires was getting an earth from the alternator cover or rubbing off the engine the battery light is really an alternator light warning you that the alternator is not charging the battery
No, but it will cause your check engine light to come on.
There are pages and pages of things that can cause a check engine light. You need to have it checked with a scantool. Then the code can be diagnosed and repaired.
You can clear the code with a scan tool or by unhooking the battery. If the cause is not repaired, the light will return.
disconnect the battery or take it to autozone If you do not have the cause of the light fixed, it will come back on.
You need to repair the cause of the code that turned the light on. You can then reset the light with a scan tool or a battery disconnect.
If your engine is running , and the battery light comes on , that indicates that your battery is not being charged
You have to repair the cause of the computer turning the light on. Then you can turn the light off with a scan tool, or with a battery disconnect.
Replacing the battery was not likely the cause of your check engine light. Most auto parts stores have a code reader will read the code for you at no charge and reset the light.