Can This Desktop Handle Starcraft 2, Team Fortress 2, Minecraft, etc without lag?

The crap Integrated video won't cut it. However something like a GeForce GTS 450 or Radeon 5670 will do quite nicely for the games you are trying to place. Either of those can be had for less than $100. A quality 450-500 watt PSU will set you back $50 or $60 or so.