You're going too overboard for somebody who is doing the most minimal of tasks. All this person needs at most is an i3 4130, and I mean at the most which is stretching it. You could even do this with a 50 dollar Haswell Celeron and a very cheap Asus H81 board.
For a case this is the most "fancy" I would do unless you decide to go mITX which would just be more costly.
Antec NEW SOLUTION SERIES VSK-3000 Black SGCC steel MicroATX Mini Tower Computer Case - Newegg.com
To answer your original question, if you get that FX CPU you'll need to buy a GPU. If you get an i3 it comes with an IGP.
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