While yes this build will work I suggest some changes:
1. Get a good power supply, while at $19 that power supply may seem like a good deal, it is of low quality and prone to failure. When it does fail it can also damage your components, so I highly suggest spending a bit more on a quality power supply. I suggest Seasonic, Corsair, Antec, or XFX (they have a very nice budget power supply).
2. Don't take a dedicated sound card as it is not necessary, and chances are that the one already built into your motherboard is better anyways.
3. If you have some extra room in your budget I would suggest a better graphics card although that one will work for what you are trying to do. For roughly the same price you could get a better Amd Hd7850 2GB: http://www.newegg.com/Product/Product.aspx?Item=N82E16814150641
However currently the best value out there is the Nvidia gtx760, for $249 it can compete with a hd7950.
4. The stock intel cooler should be fine as you are not overclocking, otherwise I would get a Cooler master hyper 212, however that is not necessary.
Hope this helps!