If central banks sell their gold in the US and buy new gold abroad (to save themselves physically transporting the gold across the ocean[0]), I suspect that the demand this creates for gold abroad will ultimately cause US gold exports to rise, too.
Selling it in place neatly sidesteps suspicion about how secure the NY gold really is and if it's all really still there, too.
If this suspicion existed then you'd see a spread between (gold price in the US) and (gold price outside the US)
When Trump screwed with the tariffs we saw exactly that. Arbitrarily messing with tariffs is one of the risks that causes people to think keeping gold in the US is not entirely safe.
Not necessarily. If relations are good, people are willing to look the other way a lot.
Either way, you aren't left holding the bag, you're giving the bag to speculators.
Do they count as export if the owner doesn't change?
I don't know. Does it count as export if it's not moved an inch?