Answer to "Notebook Vaio: How can I be sure it's the motherboard and not the cpu?"I know this seems pretty basic but have you properly discharged the MOBO without it hooked up to a power source? (Pressing and holding power tact switch for 45 seconds then letting the board sit for 20 minutes) Its a basic thing but I've actually had success with 2 or 3 systems that appeared to be dead with a simple procedure like this.
Answer to "Is it possible to change/upgrade my Graphic Card on my PC?"Technically yes, But that would involve unsoldering the current chip, and adding a new chip by reballing it on, which if you dont have the tools is literally impossible. I have the tools to do it at my shop (We do PC and game console repairs) and I still have only seen it done successfully 4 or 5 times
Answer to "Notebook Vaio: How can I be sure it's the motherboard and not the cpu?"It literally takes ALOT to kill a CPU, they dont just go bad. Have you tested your DC Jack instead of the power supply? I work at a repair shop and frequently experience a bad DC jack or the I/O chip of the motherboard going bad. It could even be you tack switch that powers on your MOBO, If you have a multimeter you can test various parts of the MOBO after watching a few youtube videos to be sure the MOBO is in fact dead.