Logitech wireless keyboard / mouse hangs

Given the new details, this looks to be a case of USB 3 interference in the 2.4 GHz band.

From Intel's note on USB 3.0 Radio Frequency Interference on 2.4 GHz Devices (page links to the full PDF download):

The purpose of this document is to create an awareness of radio frequency interference to wireless devices operating in the 2.4 GHz ISM band as a result of certain USB 3.0 devices and cables. [...] The 2.4 GHz ISM band is a widely used unlicensed radio frequency band for devices such as wireless routers, as well as wireless PC peripherals such as a mouse or keyboard.

Among recommendations to mitigate the issue, posted in a Logitech wiki at Potential USB 3.0 interference:

• If your PC has multiple USB 2.0 connectors available, separate your USB 3.0 and USB 2.0 receivers by as much distance as possible. For example, if your PC has a choice of USB 2.0 connectors, use the one on the opposite side of the PC from the USB 3.0 connector.

• Position your USB 2.0 receiver as close as possible to your wireless peripheral (mouse, keyboard, headphones, etc.).

• Use a standard, USB-adapter cable to position your USB 2.0 wireless receiver as far away as possible from your USB 3.0 connector.

The last one might be the best one to try in the case of a docking station, where space between USB connectors is usually tight.

USB 3.0 has been interfering. I disconnected the mass storage device adapter connected to the USB 3.0 back plane and now I have no problems with my wireless keyboard and mouse.