"もし返送されてきた時に今のビザが切れてしまっていたら不法滞在になるんでしょうか?
それとも、返送されてきたその日にまた郵送すれば、不法滞在にはならないんでしょうか?"
I think they will return your application you sent to them.
You should fix that and send it back to them again.
I’m sure you’ll be okay as far as you keep the receipt with you
that you paid $75(the application fee).
That’s the proof you’re still waiting for the responce from the immeigration.
Because I was ok.
In my case, the responce from them arrived
after my visa(first 6 month) had expired.
I think nobody will ask you about your stay here,
but if somebody questions about it,
you should show the receipt. Then, no problem.
(I showed my passport to Police because something happened
while I was waiting for the responce.
But it was okay and they didn’t question about my stay.)
No problem! Good luck!