Dell Studio 15 shuts down occasionally

I have a ~3 year old Dell studio 1558 that crashes from time to time. Ever since I connected my HDD to my laptop via eSata I am having problems with my laptop. While the HDD was connected the laptop crashed and since then my laptop crashes occasionally. While this problem started while the HDD was connected via eSata I am not sure if the problem is related to it. I do know that it most likely is a hardware problem. These are the occasions that the laptop shuts down on its own. It's like you pull the charger out of the laptop while there is no battery in it.

• When I watch video's on youtube. This occurs both on Chrome as on IE.
• When I stress the laptop too much, not sure if its CPU or videocard related (that is what I think). For example while browsing with Chrome (I use a lot of plugins). Also, when I watch a movie from the HDD.
• After the laptop has crashed it sometimes shuts down automatically while Windows has not even booted yet.
• Sometimes after a crash I have to press the power on button for 30 seconds, while the charger and battery are disconnected, before I can turn on the laptop again. If I don't, then the laptop turns on for a second and turns off automatically.

My own thoughts:

The fans still work, so I don't think the problem is related to overheat. The only exception I can think of is that the thermal paste is not conducting the heat properly. I suspect its the videocard, because the crashes usually occur when I watch a movie both online or local. But then again, sometimes I only browse (perhaps some Chrome plugins rely on videocard). I also doesn't really explain while it sometimes turns off one second after I turned it on.

Specifications:

Removed optic drive for a SSD drive. Also, I used a non-Dell 9 cell battery.

210-32669 Studio 1558 : Intel i5-450M(2.4GHz) Mobile CPU 1 720,31 720,31 S

230-11022 Display : 15.6 Widescreen Full High Definition (1920X1080) WLED with TrueLife 1 S

319-10317 Camera : Integrated 2.0 Mega Pixel Camera 1 S

370-15860 Memory : 4096MB (2x2048) 1333MHz DDR3 Dual Channel 1 S

400-18177 Hard Drive : 320GB Serial ATA (7200RPM) 1 S

450-14482 Power Supply : Inspiron 90W AC Adapter 1 S

490-11975 Graphics : 1 GB ATI Mobility Radeon HD 5470 1 S

555-12738 Wireless : Wireless Label CI5 1 S

555-12598 Wireless : Europe Dell Bluetooth 365 Card 1 S

555-12844 Wireless : European Intel Wireless LAN 6200 2x2 (802.11a/b/g/n) Card 1 S