ARC DECRA eligibility exemption: how to demonstrate unemployment?
To demonstrate unemployment for an ARC DECRA eligibility exemption you must provide a justification statement. 

"Justification Statement: In no more than 3000 characters, please provide justification outlining the special circumstances in support of the Eligibility Exemption Request being submitted."
- Eligibility Exemption Request: Instructions to Applicants (491 KB PDF), pg 2.

This justification does not require details of why you were unemployed. It requires you to state that you were unemployed, that you were not able to undertake research during that period, and delineate how long that period of unemployment lasted.

That is, the ARC wants to know that you were unable to undertake research for the required time. It doesn't need to make a value judgement about why you were unemployed. You do not need to explain your personal reasons for being unemployed.

In answer to your second question: No, you do not have to write the whole DECRA grant application before knowing whether you are eligible or not. As the ARC Important Dates page says, eligibility exemption requests will open and close in January 2016. You will have an answer shortly after that, about two months before the ARC DECRA closing date on 22 March 2016.

However, as someone who helps applicants develop their applications, I would always advise an applicant to draft their application well before January. Leaving it to the last two months doesn't provide much time to write, review and revise your application.