Deoxys's stats will change when traded; each game has different base stat values and movepools for Deoxys, which are recalculated when traded. Remember RBY's Time Capsule? This works the same way. Also note, that unless the Deoxys was caught at Birth Island, you CANNOT trade it.
As for the dogs, they only appear on mainland Kanto, not the islands 1-7. I've made a post in this thread
that covers a lot of the points for dog hunting; you can read up on that. Quite a few other people have made good points as well.