Unmountable_Boot_Volume is part of a notification that Windows could not properly access the hard drive. There are numerous reasons why the OS would start to boot and then produce this message, or why after Windows had been running you were shown a BSOD with this message. For example, your hard drive could be failing... or the connection to the hard drive could be faulty... or you could have had an infection on the system that interfered with drive access... or the file system could have been corrupted to the point that Windows could not read from it properly... and that's just four possibilities.
Now, using the VAIO recovery center to reinstall Windows (assuming you did not just then restore backed up programs and installers that carried infections... or that you did not then proceed to download infected software again via bittorrent) would have effectively removed any potential infection. Providing that you did not reinfect your system, if you had received that notification due to an infection, you should not have seen that issue again.
The issue appeared to have continued, however. This would indicate that the problems you face are related to the hard drive itself. This is not a conclusive indication, for the issue could be one of the hard drive controller on the motherboard as well, albeit at a lower probability.
If you wish to avoid purchasing a new hard drive blindly, I would suggest this course of action. Download a LiveCD image of some flavor of Linux. I recommend Linux Mint, as in my experience the hardware support is excellent. Burn a DVD from the image, and boot to the disc. Then, once it loads, don't install it on the hard drive. Instead, surf to Youtube, and watch some videos. Since you are booting to the OS from your optical drive and running the OS in Ram, you will not be accessing your hard drive at all... and if your issues are related to your hard drive, you should not experience them. Cost of diagnostic? One blank DVD. I also recommend you download and burn the disc from a different computer.
If your issues were indeed related to the NVidia 410M (very low probability), or Ram, then they would persist even after you removed the hard drive from the equation (As booting to a LiveCD essentially does that).
In Short... is it the Video Card? Most likely it is not the integrated video controller on the motherboard.