1. Nowadays this is not that easy, the police is not Federal anymore. The county police ask for an invoice under your name, a lease contract, etc.
3. Now as the spouse of an Argentine, you are in possession of citizenship, so, you should not worry about to extend the visa. You have to decide if you go for second class citizenship (Legal residency) or full citizenship that you can apply at a Federal Court.
You can apply for citizenship without the local green card (residency legal). It is always better to be a citizen.