Chances are it'd be too heavy. It should have a number stamped on it (the weight), I'm sure it would help if you could post it, as the axe's profile will greatly vary the weight., irrespective of length and width.
A better bet would probably be to pick up an axe from a hardware store and redo the bevels (British Red did a great walkthrough of regrinding and polish axe bevels), so the head weight is on the optimum handle length.
As for handle material, I'm not sure if birch is ok, but Hickory is tried and tested and very easily available, so I'd just opt for that; just be sure to select one with a good straight grain.
All will rise again for a better day; earth, green, with waterfalls where eagles hunt their prey