DNA is not made of amino acids. And no, not all beings share the same set of amino acids. As for sharing the 4 nucleotids, these are operational "bricks". With these 4 bricks you can make gazillions of different unicellular beings all potentialy combining and at the origin of who we are today, not just one. I suggest that the meeting of sperm and egg tells about the capability of some cells or proto-cells to combine each other and build on top of each other, in the manner of the building blocks of knowledge and information.
After all, don't we process information softwarewise through building blocks of learning? I am sure hardwarewise it follows the same path. Hence it is not just adapatation, but combinations which allow the preprograming of adaptations in new environments.
Again, does the wheel common in many devices mean that the bottle shares a common origin with the car? The car has wheels because it cannot function without them. The bottle is round because the tool that made it was round too, the blow pipe itself, that is. It's not because 2 objects share a mathematical concept or basic building blocks that they have necessary the same physical or design origin. Some things happen to be shaped thus because by pure chance the tools manipulating them gave them their shape naturaly, other things share a shape because by design they had to be shaped that way. Even beetles push wheels and balls of clay.