A build centered around an AMD A10 should get the gamecube games - not positive on the PS2 games. However, that's a kinda silly low budget for a computer that can play games decently as well as streaming media.
Also, don't put it in a cabinet: even if the case itself has good airflow, if it's venting heat into a box... you can see where that's going.
$450 is too low to only play emulators on? It's not for PC gaming what-so-ever as I already have a rig for that. I should add I don't need a HDD either. I was under the impression that emulation is mostly CPU intensive so a good GPU wasn't needed. I see your point on not putting it in a cabinet though, I'll keep that in mind. The point was to have it be used just like any other gaming console just to only play legacy games.