If the environment being captured on a video stream becomes less complicated, does the bit rate and latency decrease?
so all that is captured in the live video stream is a red light bulb
  and a black background


No. You can't get black background. You will get any possible type of noises, many shadows and all kinds artifacts of camera's matrix.


  will the new bitrate be less than X?



No. As GoPro writes constant bitrate (CBR)
No. As new noises can require even more bitrate than static image.
Yes, if you will fill background with black color in video editing software and select non-CBR mode for compression



  Will the new latency be less than Y ms?


No. Delay causes by some buffers. Buffers for N frames, buffers for N bytes. If you lower bitrate, you can INCREASE delay in some cases (actually, it's true to very low bitrates).