I believe it is Doduo or Dodrio
I have a knack for curing sleeping problems
Both have the Early Bird ability, allowing them to wake easily from sleep
Despite having sleeping problems of my own
Several pokedex entries indicate that while one head sleeps, the other stays awake, for Doduo at least. I assume the same holds true for Dodrio.
Which only leads to confusion.
I am unsure to the meaning of this line.
Dont ruffle my feathers
I believe this refers to a bird pokemon or flying type.
You'll only succeed in losing the edge
A referance to the line above, perhaps, means the ability to learn Rage naturally. Also has the capacity to learn Double-Edge via TM in red/blue/yellow.
Unless I swagger away.
They can learn Swagger via TM in gold/silver.