Why do all Dragon Slayers have Exceeds? As we see in the series, all Dragon Slayers (not God Slayers) who are trained by Dragons have Exceeds as partners. 

Specifically there are 5 of them: Natsu, Gajeel, Wendy, Sting and Rogue. Now, Laxus is also a Dragon Slayer, but he was not trained by a Dragon and became a Dragon Slayer by Lacrima. Is this a coincidence? Or is there a reason behind it?
If we follow the story of Fairy Tail from the beginning, I think it goes like this. At first, Happy was the only Exceed. Mashima-sensei created Happy to be some-kind of sidekick to Natsu. And then there's Gajeel. At that time Mashima-sensei still has no idea about the Edolas Arc, so Gajeel was on his own, even when he revealed that Laxus' thunder magic was in fact a dragon slayer magic. Then there's Wendy, another dragon slayer. Wendy is the first female dragon slayer, and since Happy was the only flying cat (the term Exceed was not revealed at that time), he decided to make another flying cat, named Charle to accompany her.

It was during this that Mashima-sensei got the idea to make Exceed as the official sidekick of Dragon Slayers. That's why he then make Gajeel wonder why he is the only Dragon Slayer without a cat. And after that, Exceed is the official sidekick of Dragon Slayers. So when he created Sting and Rogue, he created Frosche and Lector too.

Edit: The Exceed is sidekick of dragon slayers that was raised by dragons, that is, a real dragon slayer, not the one made by embedding lacrima such as Laxus (Thunder Dragon Slayer) and Cobra (Poison Dragon Slayer).