Is the assumption that the sun is infinitely far away wrong?
Depends on the answer, it could have multiple solution.
No clue.
Clear water are very smooth. Should be < 0.1 which is the current situation. Not to mention defining minimum roughness would affect other light reflection, not just the sun.
Possibly. However, sampling the sun as multiple (i.e. not just a single dot) light source has been tried, and resulted in terrible performance. Also, the multiple light source paradigm might break when a shadow map is later generated from a single POV.
Tricky problem.
Depends on the answer, it could have multiple solution.