Let me start off by saying that I would definitely recommend an aftermarket cooler for any of these CPU's, but especially the A8-3870K. I wouldn't mind giving this one a try... Scythe SCKC-2000 140mm Grand Kama Cross
only $38 shipped & comes with a 140mm fan for low noise & good airflow, but not sure it will fit in that case - Dimensions: 6.97x5.51x5.39 in I know the 6.97in is the width, & based on the pic's I'd guess that 5.51in is the height but don't know if that includes the fan or not. If it is with fan, no problem, but if that's without fan included, then the fan would be very close to, if not just touching the side of the case as it's 0.98" tall.
If you're going to give it away after your finished, I would definitely consider the A8-3870K it's a fairly powerful CPU & GPU, and it's very efficient, maxing out at just 150Watts at stock 3.0GHz in my final configuration.
I just built my mom an A8-3870K based system for Mothers Day. You can read about it here.
Very nice system, the built in GPU can handle games fairly well. I played the Diablo III BETA at 1280x1024 with maxed graphics with only a couple of slowdowns. With medium level in-game graphics options selected no slow downs. Of course it's not going to handle top tier games very well, especially at resolutions over 1280x1024, but the on chip graphics is currently the best option on the market without installing a video card.
Overclocking, I was able to increase the multiplier to 35 and boot into Windows 8 at 3.5GHz, however, that was with the retail cooler installed & temps quickly surpassed 60C & I shut it down right away as I don't ever like to see temps over 55C on a CPU.
For overclocking, your best choices are the "K"s & the 8120, they're all unlocked, meaning the core multiplier is unlocked and can be increased or decreased, where-as with a locked CPU, you can only decrease the multiplier. Out of the 3 the 8120 would likely be the most overclockable which is aparently able to reach 5GHz while playing Just Cause 2 with just a Corsair H100 CPU cooler... http://www.youtube.com/watch?v=wrvyvBJkl-c
Don't know if that was faked or not, but if it's real, I'm friggin impressed.
Just in case you or anyone else reading is lost, On the A8-3870K for example, the CPU operates at 3GHz by default. This is achieved by a multiplier & a front side bus(FSB) speed. On this CPU the multiplier is 30 and the FSB is 100MHz - 30 times 100Mhz = 3,000MHz or 3GHz. I mentioned that I increased the multiplier to 35 to get 3.5GHz, with a "Locked" CPU you would have to increase the FSB to 117 to reach 3.5GHz (30x117 = 3,510MHz) which may not be possible with some motherboards.
EDIT: oh, and if you do go with an FM1 processor, I really like the Gigabyte GA-A75M-D2H that I used for mom's computer... but the ASUS F1 offerings have a "GPU Boost" built in that they claim can increase the APU's graphics performance by up to 23%. If graphics performance was a consideration than you might consider the ASUS F1A75-M PRO/CSM