Firstly, this build is for my girlfriend, who has a small desk and wants a tiny PC to keep on her desk. The case has to be the one used or a smaller one. Secondly, it is meant to be a build that can play a few games as well as be able to record and stream Counter Strike (Global Offensive). That's probably the most intensive task the PC will be doing. I'm guessing that should be easy because of the fact that the GPU is a GTX 750 Ti, so the GPU can handle the recording (correct me if I'm wrong, but I heard shadowplay takes the load of the CPU by a lot)
For the activities you purposed, here are my two solutions.
Using a CPU+ram solution: http://uk.pcpartpicker.com/user/bd83/saved/Xy8bt6
Using video card solution: http://uk.pcpartpicker.com/user/bd83/saved/BLpzK8
I did not include a cooler, because there is no necessity for one, the stock ones will work. However, you still have enough to add an air cooler. Both will do what you want significantly better than the build you had. I'd recommend the CPU+ram solution as it'll allow for further expansion later on, but either one will do it.
Heck, MSI and ASRock have decent options, and will also give you a decent overclock. I have a MSI Z77ia-e53 and have a i5-3570k # 4.2 with little to no effort.
This is my girlfriend we're talking about, I doubt she's gonna overclock :P Like in all honesty, she just won't, I guarantee you that. Unless I push her to do so. I'll show you the build I came up with eventually. I also think AMD is a bit buggy with CS:GO I don't know man. My CPU's fans used to run insanely loud (stock ones) while running GO, and it's not even that intensive. Or even when emulating a DS game. Now the fans seem fine with GO but I'm guessing it's all software related. I mean, surely there's a reason all the pros and streamers use i7's instead of the fx series, despite the obvious benefits to playing + voip + browser + streaming (8 cores vs 4). Also ShadowPlay is very relevant to this (I believe) and so I'll post a link to the completed build. It has to include the wireless USB adapter as well as that specific case (CM elite 110). Went for the 430W semi-modular psu due to size. She has a small desk and really really likes the size and look of the case :)
Edit: Here's the build. http://uk.pcpartpicker.com/p/KqfkHx
I went for the i5 4440 as it's just a little lower clock speed but pretty decent for the money. If this was just a gaming rig and if CS:GO didn't have AMD issues I'd definitely go for the Athlon 760k but I experience first hand AMD issues :P (I have a 760k) The motherboard was also pretty cheap. 4 GB of RAM seems to be enough even for recording/streaming + playing CS:GO, which I mentioned, is going to be the most taxing thing she'll be doing. Went for an SSD instead of a hard drive as she only uses about 30 gb of space in her PC and she wants the snappy-ness of an SSD (she currently owns a Macbook Pro). The Asus GTX 750 Ti has better cooling so I replaced it with the EVGA. It's also cheaper and I'm expecting higher clock speeds. It looks better too :P I couldn't find the case and Wi-Fi adapter in the parts list so I added them as "custom parts". All of this fit into the lovely budget of £450.
GTX 750 ti isn't very powerful. 4GB of RAM is also a bit low for gaming. The Kingston SSDNow V300 also has changed over the past couple months. Kingston now uses asynchronous flash, resulting in much, much slower SSD performance(usually less than 1/2 the performance of regular synchronous flash SSDs). Since you prefer an Intel CPU, I'd suggest an i3 at this budget.
http://uk.pcpartpicker.com/p/Vg7ZXL
Add the WiFi card in. I find that USB WiFi cards are unreliable with gaming, though. You might want to look into one that's integrated with a motherboard.
Sorry, screw whatever I did earlier. Here's the new build. Is it awesome or is it awesome?
http://uk.pcpartpicker.com/p/hz3dGX
Convinced her to go mATX :D Went for the 4 gb because it's upgradable and I've mentioned this several times, she's not going to be playing battlefield or something, she doesn't even game that much :P She just likes playing a bit of Counter-Strike, and we don't need that much RAM. The CPU is just for the recording bit. I don't think the GPU is weak at all if it's giving 120+ fps in CS:GO all the time. (my 7850 gives over 140 minimum). ShadowPlay will greatly help as well. :)
My only advice is plan ahead. If the gfw starts gaming hard and planning on upgrading to 4690k and gtx 780. You want a bigger psu. 600cx would be the only thing i change.
The problem with Micro ATX on AM3+ is that the boards are very old and weren't meant to run power hungry 8 core CPUs. I wouldn't expect that motherboard to last very long with that CPU. You need to go ATX with any 8 core CPU. Consider a FX-6300 if you really want micro ATX on the AM3+ platform.
Also, the source engine(CS:GO, TF2, GMod, etc) isn't multithreaded very well. An Intel i3 or i5 would perform a little bit better(although the FX-8320 should work just fine).
HD 7850 = R7 265. GTX 750 ti at stock speeds is about the speed of an R7 260X, which is quite a bit slower than a R7 265.