Does each Kaioshin have their own God of Destruction? In Dragon Ball Super, it is told that Gods of Destruction and Supreme Kais come in pairs, so if a Supreme Kai dies, Beerus also dies. What about the other Kaioshin? West Kaioshin, North Kaioshin, South Kaioshin , Grand Supreme Kai, Elder Kai, Gowasu, Zamasu, shouldn't they have or get Gods of Destruction? 

And what about Kaioshin students who later on become Kaioshins, do they have a God of Destruction when they are born or when they are upgraded to Kaioshins? How does it work?
Supreme Kais are chosen from the regular Kais, for a single universe. One of the 7th Universe's Supreme Kais is the East Supreme Kai. This is process is elaborated in Chapter 16 of Dragon Ball Chou, when Zamasu undergoes training to become the Supreme Kai of the 10th Universe.

There are 12 Gods of Destruction, for each universe. It means that in one universe, there can only be one God Of Destruction. 

From the wikia article on Supreme Kai:


  Normally there are three Supreme Kai in each Universe, with two on duty and if one of the Supreme Kai was to die in an accident, then the currently-inactive third Supreme Kai would grow in the Sacred World of the Kai like a plant.


Therefore, each God of Destruction is linked with the Supreme Kais of that universe.

But the article also states that:


  [Supreme Kais] are the Gods of Creation who provide the catalyst for life and planets to be born, as opposed to the Gods of Destruction who destroy life and planets, which maintains the balance of the universe.


This states that a God of Destruction and their Supreme Kais have a yin-yang relationship. One cannot exist without the other.

So the other normal Kai aren't related to the God of Destruction's well being.