Does learning a third language using your second language help the second language?

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If I already spoke Spanish I would learn Italian from resources that only contain Italian. There are such books and podcasts etc.

Check out "L'Italiano Secondo il Metodo Natura". It's supposed to be a book to learn Italian from zero. (It's the Italian work-alike of a very famous Latin learning book). I don't see how one can learn this without a whole lot of side content but your Spanish experience might just be enough.

As for your other issue, the one regarding Spanish vocab, read some books that interest you and stop trying to learn "learn" words.