Screw & Glue Subfloor and Screw Underlayment

I don't have any experience to say for sure one way or the other (except that I've not had any notable vibration problems with my front load HE washer on the first floor, which is just sheet vinyl on 3/4" plywood, near the end of the joists). But what I'd think you want is basically the same thing that that you want when you're soundproofing a floor; you don't want vibrations at the top to transmit elsewhere.

When soundproofing, what you really want is a sandwich of plywood, underlayment, plywood. So roughly what you're suggesting. Cork can work well, though rubber underlayments generally perform much better at the same thickness. You lose a lot of the benefit if you screw the top layer to the lower layer, though - you want the layers glued together, without anything hard bridging across the soft layer.

1. Your current plywood is suitable for hardwood installation. I do not ever like to use 1/4 subfloor as it is brittle and when screwed sometimes doesn't lay flat. If I added to your subfloor I would at least put in 1/2 inch but like I said you don't need it with 16" OC. Note that a thicker sub might have benefits but at the same time you are adding weight to the joists.

2. Depends on area of country and moisture levels. CDX is fine in most places.

3. I would not try to match the current subfloor. I would do the opposite.

4. I do not ever attach additional subfloor layers to joists. When the joists swells - and they will in 80% of the world - your sub will be pulled or pushed and then you will have a screw or nail that is working its way out and possibly squeaky floors. The whole reason for the second layer is extra inflection levels. By attaching to the same thing you get the same thing just thicker.