Linux is certainly capable of going onto almost any machine; just without a nVidia GPU, you'll find generally X is less stable (particularly with composited environments like XGL), and driver support is less readily available. Nonetheless, with a 32MB graphics card in the old machine you won't be bothering with composited environments anyway.
nVidia looks after its Linux community pretty well.

I agree that you don't really need a new monitor. If your old one works fine there's no reason to switch (unless it's really crappy. ).

The machine you have picked out there is pretty decent and I don't think you'll have any trouble with it. The weakest link would be the graphics card, but that's not too much of a problem if you don't do a lot of gaming/3d stuff. The GPU will be able to handle Oblivion just fine, so there's a 'benchmark' for you.