DIY 120fps video recording and instant replay
I believe the answer is no, no, and no, for several reasons.

First off, GoPro don't support their cameras as broadcast cameras.  They are person video recording devices.  I tried using the HDMI output of a Hero3+ camera to feed into a studio broadcast system, and when it didn't work the response was "it's not supposed to".  The Hero3+ (and I presume the Hero4) do have ways of sending video over wifi so that you can get a vague idea of what the camera sees, but it's nothing like full resolution.  So the first "no" is that you cannot really extract good video from the GoPro in real-time.  You have to record it and them dump the card.

The second "no" is that 120fps is just barely starting to be supported at the video transport layer.  Yes, people have been talking about gaming consoles that can achieve 120+ fps for some time, but actually driving that kind of speed to a monitor?  Very hard.  And impossible with the HDMI output of the Hero4, which is limited to 60p: https://gopro.com/support/articles/hero4-hdmi-output-information

The third "no" is getting that 120fps input into your computer.  There is screen recording software that can scrape 120fps output from your GPU, but recording 120fps of 720p (or greater) is still a very specialized thing that's outside your "cheap recording and instant replay system" parameters.  There are video capture cards that claim 120fps recording, but they are cheating: they are multiplexing 4 SD signals (thus really recording 4x 30 fps and calling it 1x 120 fps).