How did Shinigami come into existence? How is it that Shinigami come into being? If there was no way for them to come into existence(in the human world), would they just slowly die away when any of them kill for love? Also, can Shinigami die of old age?
Shinigami, or death gods, rule the world of Death Note. A shinigami's purpose in life is to kill humans to extend their own lifespan, through the use of a death note.

Well, as you have already noticed, there are many different definitions of shinigami. The word is usually translated as "god of death", so you can think of them like grim reapers. And since the word "grim reaper" is also used for a variety of things, you can see how this can get interesting. There is no set definition for what a shinigami is, so the writers of different shows can just make things up. For your story, you get to decide whether you have to die to become a shinigami, and you get to decide whether certain people become shinigami.

In Mobile Suit Gundam Wing, the word Shinigami is used to refer to a character whose friends and family kept dying. It's a really loose word.

But if you're asking specifically about Death Note.... In Death Note, the shinigami do not have to die and they are not people who have become shinigami. The manga doesn't really go into the history of the shinigami, but it is implied that they have been shinigami for the entirety of their existence. 

One of many stories about Shinigami coming to existence is :


  
    The Shinigamis become how they are because they commited suicide and as punishment they have to collect souls.
    If a shinigami fully remembers his or her past life before becoming a full shinigami, he or she disappears and becomes a ghost. They become full shinigamis by collecting their first soul.
    Personally, I think this is more easy to understand.