Must total world net exports be zero?
That depends on your model of the world, and on how you define "goods and services". 

In a gold standard world, you essentially exchange goods and services with gold. Gold mining is therefore simply an expansion of the money supply. Adding gold to both the nominal and real side of the economy seems to me like today adding money as a good to GDP. If you do this, you get a positive trade balance. It makes a lot of sense to seperate monetary gold, which is on the monetary side, and commodity gold, on the goods side, as Henry mentioned in the comments.

In my opinion, it does not make sense to count gold twice. It seems reasonable to seperate the monetary and real side of the world, which then would yield a balanced world trade.

To the second question: I am no expert on gold mining, but I could imagine that it is not possible to "drive up" gold production in a recession. What is true is that a (nearly) constant money/gold supply leads to inflation being counter cyclical.