my charging cable overheated while charging my airpod case now my airpod case is not charging at all. What should I do?

Can you try this?

Full re-pair, reset:

Settings > Bluetooth > Airpods > Forget this device. Confirm it.

Put both Airpods in their case, charge them via Lightning for a few minutes (or preferably until the case is 100%). Unplug them. Open the case and hold the button on the back of the case for 15+ seconds or until the light flashes white. Hold them near your iPhone and see if the pairing screen comes up and pair them. If not, repeat until it does. But if you can't see the screen pop up, then manually pair within Bluetooth.

If it doesn't work, open a case with Apple Support here: https://support.apple.com/airpods

Also if you can see it, please tell me the firmware ver.

Settings > General > About > AirPods > Firmware Version

Yeah if that's the case, you'll have to open a case with Apple Support here: https://support.apple.com/airpods

Good luck

The cable... itself? Overheated? How do you know this?

Try another cable. If it doesn't work, open a case with Apple Support here: https://support.apple.com/airpods

Do you have another Lightning cable that you can try?? Or try plugging that same one into your computer's USB port and try to charge your AirPods that way.

But does the case respond with a white light when you press the back button for 15+ seconds?

You can connect them to your PC or anything that has Bluetooth.