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

I mean if you have a usb lying around, putting ubuntu on it and trying is free. Worth a shot, you could get access to the drive too if it works. If that doesn't work you could still bring it into the shop to get worked on.

Open that bad boy up, it needs some hardware inspection. Check for water damage, dust accumulation, zapped capacitors, loose screws and parts. Try to get a command prompt going and use the tools like others said.

If reinstalling windows didn't fix the issue then you most likely have a hardware issue. You could always make a linux live USB and see if that works. The live USB won't touch your HDD so if the HDD is the problem you won't know until after you've installed linux.

Before you do that download a program like hwinfo and look at your temps. Specifically CPU temp. Somewhere towards the bottom it will tell you stats about your HDD as well and it may include what SMART thinks your HDD's health is.

Try a live USB distro first. If it is your hard drive It will eliminate that as a problem temporarily letting you diagnose.

If the windows reinstall won't run or won't complete, It could be anything from a dead fan to a bad memory controller.

I have no idea why.

Sounds like HDD failure if its less than 2 years old it should all covered under warranty.

What laptop vendor? Most vendors have a hardware tester that you put on a usb and test for hardware issues.

Same for windows. You can download windows usb creator utility and use it to reinstall windows from scratch.

I am a complete amateur, so take it with a big grain of salt, but it might be hardware problem, maybe the SSD has gone bad. The best way to test it might be to try install Linux, and use it for few hours from the USB itself in live environment

You should look at event viewer to see what is causing the problems. If the problem is overheating, a bad HDD, or bad ram then I doubt Linux would help.

First I'll say, if you aren't able to boot a windows installer off of a flash drive (and nothing's wrong with your flash drive and you created it without errors and there isn't a setting in bios preventing you from booting it like disabled USB boot) then it could be a hardware issue that Linux won't fix (it's not clear how you tried to reinstall windows). But if you're able to get to the windows logo or the menu for safe mode then it sounds like it "POSTs" (gets past showing the bios screen) and windows should be reinstallable. In short, I think it's unlikely to be a problem that only Linux can fix if you want to stick with windows.

That said, if you are otherwise interested in trying Linux and create a USB installer, most will allow you to exit the installer or choose to go into a temporary "live desktop" where nothing is saved, so it's a good opportunity to try out how that distro feels to use - just don't save anything important. And if you do end up installing either OS, you can use that "live mode" to use programs to back up any files from your main hard drive to another plugged in drive.