I described the scene of him drive away.
I described the scene of him/his driving away."
That said, I would write that sentence like this:
I described the scene in which he drives away.
driving away is a gerund phrase, in the prepositional phrase introduced by of.
a gerund phrase is basically a type of noun phrase.
I think you are confused about what the subjects of your example sentences are.
1) Behind me is/are cute puppies.
The subject here is "puppies," not "behind me." This sentence is equivalent to:
1*) Cute puppies are behind me.
The subjects of the other examples are, in order, "two tall buildings" and "a scenic view." From there you should be able to ...
Hard to say which one is correct as they both mean very different things.
... at 37 minutes
This would mean 37 minutes from the start.
... 37 minutes before the curry is ready
This would mean 37 minutes before the finish. While it could be used in other contexts, it seems highly unlikely such an instruction would appear in a recipe method, as it ...