When two people are mutually prohibited from benefiting each other and one returns the other's lost article in a place where a reward is customary, where does the reward go?
Since the finder may neither keep the reward (that would be benefit for himself) nor let the owner keep it (that would be benefit for the owner), it must go to hekdesh — or in the present age, to charity.
Question 2
What is the halachic mechanism that allows two vow-bound partners to legally use city-shared spaces like a bathhouse or synagogue?
By signing over one's portion to the nasi (or another), each partner entering the bathhouse or synagogue is no longer entering the prohibited party's domain — the ownership has genuinely passed to someone else.
Question 3
Two vow-bound partners share a courtyard that is too small to divide. Which activity is forbidden to them regardless of divisibility?
Partners in a courtyard normally have the right to object to a mill, oven, or chickens. Under a benefit-prohibition, failing to exercise this right would constitute providing benefit to the other. Hence this is forbidden whether or not the courtyard can be divided.
Question 4
Why is a vow against benefiting from an entire nation treated more leniently than a vow against a single individual?
The rabbinic decree against selling (lest one come to buy) applies only for an individual vow, where every interaction with that one person carries risk. Against an entire nation, if one transaction is problematic the person can simply deal with a different member of that nation.
Question 5
A father-in-law is forbidden from benefiting his son-in-law and wants to give his daughter money for her personal use. What must he say to ensure the son-in-law acquires no rights?
Married women's property normally falls under the husband's jurisdiction. To prevent the son-in-law from acquiring rights — which would violate the vow — the father must explicitly exclude the husband's authority AND specify the purpose (food, clothing, etc.). Simply excluding the husband without specifying purpose is insufficient.