Dorna has 18 Dex, 13 Str and no Weapon Finesse and yet she can still hit gnolls with a battle axe and short sword while I struggle with my 18 Dex elf bowman. This is bizarre. Any explanations? She should be completely useless.

And I cannot remove her battle axe from her inventory and leave her with small weapons to dual-wield..

First, does she have the two weapon fighting related feats? If not, then I agree that she shouldn't even be able to hit the ground.

If so, then she still gets +1 to hit with the axe for her strength, and +4 to hit with the short sword from finesse. Those stack with her attack bonus, and any other skill or magic effects that might provide a bonus.

Is your strength at least 10? If not, then I think you'll take a penalty to attack rolls (not sure about this). Likewise, you might be taking a -4 to hit if you are shooting while in melee range of an enemy, unless you have the feat to negate it.