Oh yes, you should own the OS. If it was preinstalled in the computer, you've bought it. In case of Windows 8 in a brand computer there might not be any sticker with a code, though. Only one saying Windows 8 installed. The license code will be in the bios or whatever they call it now and it should automatically add it during the installation process if you have the right media.
Nowadays you don't get any reinstall disks, you should create them yourself after seeing the nag screen often enough - unless you're one of those who simply check the "don't remind me again". However, those disks only work with the computer they've been created with.
Since this one seems to be your only computer, get a new one, a real gaming computer this time. Send the broken one to be fixed under warranty and either use it for less demanding tasks or donate/sell it to someone who only needs it for banking and e-mailing.
I just installed a new hard drive to a laptop and did a clean install. I charged €150 including 24% of VAT. The price of the HDD was €65, Windows7 license was on the sticker in the bottom of the laptop. That's the average charge for that kind of a job here.