When turning verbs into nouns, when should -MA with possessive suffixes and the -DIK particple also with the possessive suffix be used?

Here are some examples:

Trenin kalkmasına az kadlı. Little (time) remained for the train´s departure. This uses -MA.

Sorununu çok iyi anladıklarını biliyorum. I know that they understand your problem very well. This uses -DIK. 

Why can you not say...? 

Trenin kalkdığı az kaldı.   

Sorununu çok iyi anlamaları biliyorum.

Your question is a very good one. Most Turks could not explain it in a hundred years even if they intuitively make the right choices. A learner can acchieve a slight intuition also after diving into the language deep enough, I know it from experience.

While we wait for the masters to clear it allow me to tell you my learner´s view  -  or maybe mix it a little bit more.

The verbal noun -mA denotes the somewhat abstract concept of an action or its result. -DIK participle is narrative. Your examples are good:

The first action, the train´s leaving, doesn´t exist anywhere else but as a concept in the speaker´s head. Similarly, -mA infinitive is used when the result of the action is the issue like in

                  Trenin kalkması beni biraz üzdü. ´The train´s leaving made a little sad.´

Also when we talk about actions that are necessary we see them as abstract (future) concepts in our head.

                   Gitmem lazım. ´I must go, my going is necessary.´

The place of possessive marked -DIK participle is often with verbs denoting saying, thinking or knowing. What do we say, think or know? Things that have/haven´t actually happened, actions we want to relate, to tell about, to share. Like in your example, jas206: the speaker wants to say he knows what the actual state of affairs is.