(Almost) by definition, questions found in Shut are the ultimate "on-scope" questions.
Should we populate questions with Shut (like start with Be'er Moshe [Igros Moshe forbade translations of his Shut] vol.1 siman 1 and continue)?
(I'm assuming you mean sh'elos usshuvos generally, not the linked-to Web page specifically. If you mean the latter, then I don't see what's special about that page, and my answer might be off the mark.)
Many questions in volumes of sh'elos usshuvos match the standard closure reason of
too localized: This question is unlikely to ever help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet.
(In fact, I understand that that's why R' Moshe Feinstein forbade — as the question mentions — translation of his t'shuvos.)
However, many questions in volumes of sh'elos usshuvos do not match that closure reason. Among those, some are out of copyright status. (Presence on hebrewbooks.org does not AFAIK necessarily imply a lack of copyright status. The copyright holder could have given that site, but not (e.g.) this one, permission to copy his book.) I don't see the harm in copy-pasting such copyright-free questions and answers here, and IMO there is certainly benefit in that people will have them in translation.
But note that duplicate questions should be one on the site, not split simply because they appear in different volumes.
To msh210's cautions about excessively localized questions and about copyright issues, I'd add that copying and pasting or straight translation of the question may not be appropriate in the context of this site. The original question may have included elements that would be out of context here, such as references to the rabbi to whom it was addressed or information about the asking person/organization. In addition, an answer here quoting the original answer should, naturally, attribute it to the original author.
Finally, in some cases, responsa volumes include questions that were intentionally leading questions, sometimes by the answering author himself, borne out of a desire to provide an answer rather than out of curiosity. When adapting such questions for this site, make sure to rewrite them from the point of view of someone who's genuinely curious.