How can the statement that "by the right of your Prophet or by the right of so-and-so" is forbidden because it is an oath by other than Allah be reconciled with the hadith: "I ask You by the right of those who walk towards You and by the right of those who ask You"?

The preponderant view is that asking "by the right of so and so" is not an oath, but rather a type of tawassul (seeking nearness to Allah). If the request is made by a legitimate right, such as "by the right of my prayer" or "by the right of my walking this path," then it is tawassul to Allah through good deeds, which is permissible. Sheikh al-Islam Ibn Taymiyyah holds that the "right of those who ask" and the "right of those who pray" is a right Allah has obligated upon Himself, just as one asks by faith and good deeds, which He has made a cause for the answering of supplications. As for tawassul through the Prophets, it is by believing in them and obeying them, or by their supplication and intercession, not by their essences. And according to the view that this falls under the category of an oath, it is an oath by an attribute or an action of Allah, because His answering and rewarding are among His actions and sayings.