Why did Mr Nobody keep asking for more money but just won't kill John?

When I saw the dog I thought he might have been a relation of Halle Berry's character from the previous movie. But there is no actual reason for his character to exist.

He was upping the stakes to get a higher price. In the end he changed his mind because Wick “saved” his dog.

Everyone is so fucking sentimental in this movie …

To later provide additional proof that John is a likeable person.

It's a dog thing. Also, he hates the Man. After watching the wall to wall whoop ass Wick laid down he figured out that this was the best opportunity to take the Man down, and maybe he'd pay millions of dollars to see that happen, which he effectively did by avoiding the kill shot. Also, if this guy is a position like this to negotiate 20 mill plus contracts, there's surely another high paying job just around the corner.