Could you live off $1,200 a month? Why or why not?
It's hard to say. On the face of it I'd say probably not but if it ever became essential you never know what could be trimmed from the household budget.
It would be cold in the winter because the heating would have to stay very low. I don't imagine you'd get a lot of protein in your diet for that kind of money, although shifting from meat to more vegetable based protein would reduce costs. I don't know how you'd find spare cash to fund things like consumable parts of a car on such a limited budget.
If you had to deal with paying rent or a mortgage out of it I don't know if it would be possible. In places like NJ you'd probably need you $1200/month just to pay property taxes.