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It really depends on what the cause of the check engine light is.
Yes you can. But you will want to check you're engine soon.
It is safe to drive you car for at least one mile after the check engine light comes on.
Replacing the battery should not have triggered the check engine light. The check engine light comes on when there is a problem with the emission system. Saying that, I would drive it for a while to see if the light does go out. If the light does not go out have the ECU scanned to retrieve the trouble code. Then use this code to diagnose the problem.
yes
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.
It is not recommended. A flashing flashing check engine light is an indicator of a catalytic converter damaging misfire.
yes
Definitely not, get it checked imediatly
the check engine light goes on and off
That is what the check engine light is for. It informs you of a failure.
Yes but, if the light is flashing no.