Sounds like a BCM (body Control Module) or a Ignition lock cylinder problem. You said the battery light is coming on. If it is coming on with the car running, and the alternator is charging, You probably have a faulty BCM. We see a lot of them on that year of impala. To make for sure you would have to pull codes out of the bcm and diagnosie those codes. If a BCM is at fault, It would need to be programed after replacement. So it would need to be done at a GM dealer or a shop that has the right scan tool. Let me know if this helps or if you need more information.
BCM runs about $200.00 so do Lock cylinders.
Labor from $100.-200. depends on the shop
Yes, The door locks are controlled by the BCM. With a BCM/ security issue, it can prevent the car from cranking.
Double check your battery cables, especially the positive side. Make sure there is no corrosion between the battery post and the cable.
Is it The factory battery?
Replace the cables if they are corroded. Replace the battery if it is leaking.