Could Linux save my laptop?
Depends on the root cause of the BSoD. If it's a hardware issue, then no, installing Linux won't fix an underlying hardware issue. What does the BSoD screen actually say? Any specific error code?
My understanding is that Linux is a kind of system that you download the components to a USB or what not and then install it on your machine. Is that something I could do in this case?
Yep, you could do that, but you'll need a functioning PC to create a USB installer for the Linux OS of your choice. You'll need a blank USB drive and some software to "flash" the Linux OS installer to the USB drive. e.g.: https://linuxmint-installation-guide.readthedocs.io/en/latest/burn.html
Gotcha. You mentioned you put the PC into Recovery Mode and tried to reinstall Windows, but it didn't work. What happened that prevented the re-installation of Windows? Did the PC just shutdown during installation, or were you wholly unable to get the Windows installer to run?
If you're still able to boot into Recovery Mode/Safe Mode, try opening up a Command Prompt and running the SFC Scan and DISM commands outlined here: https://www.howtogeek.com/222532/how-to-repair-corrupted-windows-system-files-with-the-sfc-and-dism-commands/