When should one use pending, as opposed to left or remaining?
"Left" and "remaining" are synonyms.
I have only four eggs left.
There are only four eggs remaining in the refrigerator.
You can also use "remaining" as an adjective, as in
The remaining eggs were rotten.
You couldn't use "left" in that construction.
"Pending" is a different matter. It refers to something that is about to happen or is in suspension for the moment.
Their pending divorce weighed heavily on the children.
It can also be a synonym for "until", as in
Pending a trip to the grocery store, we will not be able to make omelets for everyone, since we only have four eggs left.