All laptops shut down when they overheat. This is done to avoid permanent damage, etc, etc. To prevent it from further overheats in the future, I'd recommend using a can of compressed air to blow out each of the vents that your computer uses as cooling. If you want to, you can actually disassemble the entire laptop and do a thorough cleaning of the likely very dirty fan. On the software side of things, check your Task Manager when the computer is on. Take note of any processes that use excessive amounts of CPU time or consume large amounts of RAM. You may be able to uninstall the programs that do that.

If software is not the issue, you may have to replace the entire screen assembly. Don't fret yet, though, as there are some things you can try before resorting to the 'Nuclear Option'. Try cleaning the contacts between the display assembly and the motherboard. Laptops are prone to collecting massive amounts of gunk and dust over time, and swabbing the connection with liberal alcohol may fix it. Remember to dry all components before reassembling the laptop. If that doesn't work, there are other possible causes of failure. This includes power supply issues, motherboard issues, and graphics manager issues. If your computer supports it, you may be able to run an ePSA. This boots your computer into a mode where it systematically runs onboard tools to determine what may be broken. Different computers has different ways to access this mode, but you might be able to access it through booting into the BIOS mode.

All laptops shut down when they overheat. This is done to avoid permanent damage, etc, etc. To prevent it from further overheats in the future, I'd recommend using a can of compressed air to blow out each of the vents that your computer uses as cooling. If you want to, you can actually disassemble the entire laptop and do a thorough cleaning of the likely very dirty fan. On the software side of things, check your Task Manager when the computer is on. Take note of any processes that use excessive amounts of CPU time or consume large amounts of RAM. You may be able to uninstall the programs that do that.

Yes. You'll have to detach the entire upper display assembly, in most likelihood, however. You might be able to buy that directly off of ebay or something, but I don't hold hopes for being able to just replace the glass.