Can we let a language die if we record, document and translate it?

If you have to ask the question, it’s already doomed. Linguists can only record the language; only the language community can give up on it.

If the  language community wants to hold on to it, linguists can give them tools. But it’s not easy.

EDIT: OP also asks:

Also, why should we teach our childs our mother tongue when we are living abroad (in a country with another language ) and also our first language is not in danger?

Because more languages—even if they aren’t much practical use—are a good in themselves.

It doesn’t look like I’ll get to have kids; if I did, I’d teach them Greek. Because  I’m proud to be Australian, and I’m proud to be Greek. And that’s something I’d want to share with my progeny.

My parents did not speak Greek to me until we moved to Greece. In retrospect, that’s heartbreaking.