Yes. We do surface mount on an auto pick-and-place, solder and inspect, and then add the thru-hole parts by hand. Make sure the solder paste stencil doesn't paste the thru hole pads.
They will if you paste them. Don't. The flux should be cleaned up by the assembler so the holes will be clean.
ENIG (gold over nickel) PCB finish is fabulous, beautiful to solder after years of storage. Solder coating can get a bit oxidized by the reflow oven.
It's not hard. People will make laser-cut mylar paste stencils cheap, sometimes free. They won't last as long as stainless but they are fine for small quantity production.