You might be able to replace the part. The only way to really find out would be to look it up on Sonys website. See if you can see the part listed somewhere buried deep in the bowels of their support pages.
The only other thing I could suggest would be to try a different wireless network. Maybe even set your phone if you had an iPhone as a wireless point and see if it holds the connection. If it's constantly dropping out then for sure it's the laptop. You could try a format to really rule out an OS issue or simply boot from say a ubuntu disc and try connect to a wireless network from that (i'm pretty sure something like that should work). If it stays strong under something like ubuntu well then .... back to software issues!
Probably the LCD or backlight within the LCD .. So probably the LCD anyway. You could check by plugging an external monitor into the laptop and seeing if that ends up dimming and flashing or if it's stable. If it also dims and freaks out then it could actually be the GPU.
Yep. Just google search DVI cable and you can see how pretty they are! If you're using a mac then it could get a little different but since you mentioned a tower, I assume it's just as PC.
@Zero Shift
Have you tried installing the latest catalyst beta drivers from the AMD website? You can grab them here -
http://support.amd.com/us/kbarticles/Pages/AMDCatalyst1211betadriver.aspx - I'm running them and have no issues. It sounds like you've done the right things, Used driver sweeper to ditch everything it's found, It's worth trying that in safe mode BTW. The only random thing you could try would be to throw the card into another PCI-E slot and see what the heck that does.