Are go-arounds prohibited at St Barth (TFFJ)?
The video's description reads:


  After passing Eden Rock (min 02:39) it is prohibited to conduct a Go Around.


Below is at 2:39:



The official eAIP reads (link may die):


  AD 2 TFFJ.AD 2.22 Flight procedures
  
  22.2 Arrival flights:
  RWY 28:
  
  Going around is prohibited in short final.


If Eden Rock is that tiny island on the left, then it makes sense as the plane would be aligned and on short final after passing it, where a right break option is no longer available.

So the answer to the first two questions is yes.

The reasons are not on the chart, so possibly the clearance gradient was not met, was not demonstrated, or the risk to the road and community past the tiny hill is too high.

Also note AD 2 TFFJ.AD 2.21:


  Take-offs on RWY 28 prohibited.



(eAIP chart)