Texas Green,
I also live in Austin. Many of my fig trees lost their leaves and died back to ground after cold weather in previous years. In the past, I was fortunate because the trees grew back from the roots.

Probably, but only time will tell.

simmilar thing happened to me last year to 2 brown turkeys. they both came back. Even a few branches that I thought were history sprouted new growth. you should be alright.