You called Coolermaster about your Compaq battery? They're not really qualified to troubleshoot that.
Heat is damaging to a battery, but I can't say for certain overheating is what killed it. We can't determine yet that the battery is the problem. The only way you can know for certain is to purchase a replacement battery and test it. If the laptop shows the same problem, then the issue is likely the motherboard or the contacts.
Never go back to the manufacturer for a battery. Their batteries were made when the laptop first came out, and have been sitting on the shelf aging. They are sometimes DOA. They are never cheaper than $200.