Hey there, catsears.
Well I can't say for sure if the HDD is having issues or not, but it seems like the main problem is not the drive. If it was constantly overheating some of your laptop's components might have been damaged, so it would be best to take it to a computer services center or get in touch with the laptop manufacturer's customer support for more information.
As for the HDD. If you have valuable data written on it, even if you laptop does not start, you should be able to get to those files as long as the HDD is in good condition. Just connect it either externally, or internally to a different computer and try to access it. If you're having trouble with the HDD, you could give it a try with Ubuntu Live USB/DVD, to see if you'd be able to get to your information this way, or by using data recovery software: https://www.lifewire.com/free-data-recovery-software-tools-2622893.
Hope that helps. Please let me know how everything goes.
Boogieman_WD