I can't find the forum posts I read that in but I believe one was here on tomshardware and the other was on dell but like you say if the system sold with the processor it should support it. So about that video card....
Again, the system wasn't sold with Core2 it was sold with Pentium 4 Dual core aka Pentium D.
What about the card ? It's a pos that can drag an entire system down all by itself
A 5670 would be a good upgrade, you could get away with a 5770, I don't think the system is worth spending that kind of money on though .