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Also, I've seen that your website offers the functionality of showing any given user's messages. Is there an analogous way of filtering messages by board? Like: "showing Economy messages".
Nope. That's TryNinja's specialty. Again: just ask nicely 
Link you mentioned TryNinja already offer API where it's documentation can be seen on https://docs.ninjastic.space/. If OP willing to write script which download topic from the API and wait for several days, it should be viable option.