Not quite right but mostly accurate - some the water also comes from the magma and is exuded before the chamber empties. This interacts with the surface water you mention coming down in the brecciated dikes and veins, and enriching the deposit, usually while the magma chamber is itself cooling and crystallizing.
Maybe. The subducting crust is dewatered way deep in the mantle and the water lowers the melting point of the mantle rock – that’s why it starts rising in the first place.
Cite.
I’ve not been anything in the simpler sites I’ve perused that explains what happens to that water. Does it make it to the caldera zone? Is it split and form mineral oxides and hydrides? Does it take a shuttle back to the nearest ocean? 
It’s frustrating.