What is the name of this minimalist film genre?
Generally "Minimalism" is more about shedding content to focus on the essence, "eliminating all non-essential forms, features or concepts." This does not necessarily require the elimination of sound or strict use of b/w.

Another term used for a Minimalist like film back in the 1970s was "lean".

What you are describing is more like emulating a silent film which reached its peak in 1928.

If you only use b/w, 60 seconds, no camera movement (pans, tilts, tracking), no cuts, no edits what you have is one clip from a stationary camera. This in itself does not fulfill what "Minimalism" in cinema is about as the balance of this question rests with whether the content is in fact Minimalism.

Also, specifying certain aspects such as b/w and no sound are somewhat superficial now.

Consider that before 1928 that sound was generally limited to what music or sound effects could be played live with the film. So too, color films were not widely available so only hand tinted films existed. Before 1928 the cinematographer generally did not have the option of synchronized sound and color. To go back to this approach is a conscience artistic choice today where as back then, you did not have this choice cause that's all you had.

I would call the genre you are describing as approaching Neo-Silent but not quite there as these guys use hand-cranked gear etc, as described in their manifesto:

http://www.silentmovies.com/neosilent/neosilent.htm

So perhaps, "Post-Neo-Silent" is a more accurate term?

In any event what you are describing is actually very popular in some cinema circles. Consider director, Blake Whitman's "1 Minute Vimeo Project":

https://vimeo.com/groups/1minute

Wherein Blake offers:

"Welcome to the 1 Minute Vimeo Project Group. Add one video, comment on two!

With the chaos of everyday life consuming our senses, it is nice to sit back and enjoy moments and experiences that are often overlooked. This project aims to study the forgotten moments and times in life that we often pass by without acknowledging.

The rules:
-Video must be exactly one minute long
-No camera movement (no panning, tilting, etc)
-No editing whatsoever
-Use original sound
-Tag with '1 minute'

-Have fun"

Aside from allowing color and sound, this is pretty close to the criteria you have defined.