with three devices plugged into it.
I have found that for the HDMI switch to recognize another device, automatically, the other devices have to be turned off.

I got around that by using a HDMI switch with its own remote control which was soon replaced with one of those Harmony Remotes.

You didn't specify how many ports your switch has, or whether it has its own power supply, but you may find that one of those ports is the master.

HDMI should supply 55mA on the 5v supply so, theoretically, your device should be getting enough, as long as the mini meets HDMI specs

P