Yeah your family computer is good, except for that built in video card which is whats making games run slow. I'm surprised it runs Half life 2 at all!
Yes that computer is good that you are wanting, though that CPU is very expensive for only a .5Ghz boost compared to getting something like a 3500+, though its your choice.
As for the graphics card, that is very powerful, so yeah your getting a really powerful system there
I don't get how poasi gave you information though and you didn't say thanks to him as it must have took quite a while to gather, and still came up with custom specs of your own anyway.
On a good note though, that will run things for a very long time, maybe a few years, and should leave you room to update later
EDIT: poasi I think Dishdog meant the system he has now, not the custom parts you considered, and it would build a good system, and for only 500 quid!