Would a Universalist Wizard casting HoA equipped with a quarterstaff and having the TWF feat be able to do
1d6+str/1d6+str damage?
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As a standard action, you can make a single attack using a melee weapon at a range of 30 feet. This attack is treated as a ranged attack with a thrown weapon, except that you add your Intelligence modifier on the attack roll instead of your Dexterity modifier (damage still relies on Strength). This ability cannot be used to perform a combat maneuver.
no, since the ability is a standard action
no, since the ability states single attackno, since two weapon fighting is a full round action
HAnd of the Apprentice also states you make a single attack roll, so only 1 set of damage dice.
If you want a "more damaging" version of Hand of the Apprentice for a wizard, there is the sword binder Sword of the Mage with a greatsword.
Sword of the Mage also allows the delivery of touch (and ranged touch) spells through the sword at higher levels; granted, BAB is still +1/2 levels but eldritch knight might be something to look at after wizard 8.