日本語を勉強するために、日本に住む必要がありますか。 

→This one is better (^o^)

And can I use ために in this context at all?

→This is difficult question.

   I think "ために" is similar meaning with "for" and "to" of English as below;

  ・If the object is noun (for example a person), it's applied "for".

      私は母のためにプレゼントを買った。

      I've bought a preasend for my mother.

   If verb, it's "to".

      私は日本語を勉強するために日本に住んでいる。

      I live in Japan to sutydy Japanese.

I don't think above answer is complete as it'S my sentient.

So it is just for your feference.