So Aroden's Spellsword says:
"If the rod or staff can be used to make attacks, you can attack normally with the weapon or use the weapon as if it were the merged rod or staff. If the effect created by the rod or staff requires an attack roll to successfully strike a foe, you can make the attack roll as if you were making an attack with the weapon at its highest bonus (including any bonuses the weapon would normally receive) rather than just a normal attack with the rod or staff—doing so does not allow you to add the weapon’s damage to the attack, but instead allows you to use your skill with the weapon to boost your chance of hitting with the rod or staff’s attack, spell, or effect."
If you put a rod of lordly might into a plain wooden staff, would your monk flurry with a 1d8, x3 crit, +4 slashing weapon if the rod was stored as the battle axe? Would it be bludgeoning but keep the other stats or something in the middle?
No.
You are attacking with it either as a plain wooden staff or as a +4 battle axe.
If you had a +5 staff and say weapon focus (staff), you could use the +5 from the staff and the +1 from weapon focus in, for example, the touch attack for hold person.
Got it. It only affects tohit and no other property. So the primary function is really just the extra "hand" so the rod doesn't need to be stowed or retrieved.