With the A10-5700 and the BIOS fan control set to "quiet" for the CPU, ANY HSF with a 4-pin fan and a low rpm range and 3 or more heatpipes is going to be basically silent. The "peak" fan noise of an HSF is irrelevant in such an application, as it will never run at that speed as there simply will never be the cooling demand for it when aided by the awesome heat spreading properties of a heatpipe. A gentle breeze over a few small heatpipes is all that is required to keep an A10 below thermal margins at stock clocks, and in this case, being the 5700 (low TDP model) it's even less of an issue.
Almost any modern high efficiency ATX PSU will have a variable speed fan, and, again, at the dissipation of an FM2 build just isn't going to call for any more cooling than is provided by the lowest fan speed setting.
The Arctic A11 is all that is required to achieve the silent goal here. ~$23
The Seasonic G 360W is all that is required to achieve relative silence for this machine in the PSU department. At ~200W and under it runs near dead silent.
The Fractal Design Arc Mini R2 is a solid choice for a quiet system, as it includes a 3 speed fan controller for up to 3 case fans (comes with 2, which will be plenty for this application).
The arrangement of the case fans in the arc mini will be in alignment with the way the A11 mounts on the board, providing a smooth quiet current of air from the front to the rear of the case, assisted through by the HSF. Set the fans to the low or medium speed.
The loudest thing in the system will be the hard drive.