So here are my thoughts about how to implement this!
First, hooking the API up to a chatroom style website; I have no experience in that, but I think it should be relatively simple to take existing software and hook it up to the API, for anyone who has any experience in either of those things.
Second, have a page for instance admins to submit their instance, along with the account on their instance that has been created for the purpose of this hook-up. That account can be set up with whatever posting publicity level that the admin wants, be named whatever the admin wants, etc.
Third, I agree that a fully anonymous feed into mastodon is less than ideal. A team of volunteer moderators on the "Peanut Gallery" site filtering the questions is a decent solution, but lacks the immediacy of the chatroom set-up.
That flaw could be mitigated, however, if instead of having it as a chatroom per se, it has it more like... a commented FAQ? A discussion board? The questions that have been submitted and answered are displayed in a list, and clicking on one opens up a page with the responses from people on the various linked instances.
So then, instead of a chat room to ask a question, there are two primary functions on the site: "Submit a Question" and "View Q&As", the latter of which is the listing, and which could be sorted either by number of responses, number of instances which responded, or number of views.
I think that idea would require more coding than the simple anonymized chat room, but would probably be more useful overall, especially in the sense of having a browsable history.
Editing to add a follow-up: There should be a function when adding your instance to the site to have it queue up previously asked questions, so your instance gets a chance to add answers to them! This is important.