Tile – If the top layer of the subfloor is structurally sufficient, can I skip patching plumbing holes in the bottom layer

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.

I have done several tile floors by just screwing the hardie board down and then laying thinset and tile right on top. I only just heard recently that some people recommend that you should lay down thinset under your hardie board.

Since you are putting down hardie board on boards with small gaps, I would not recommend trying to put down thinset under it. It will make an enormous mess under the house (or whatever is under your bathroom). It is also more likely to make the floor uneven since it is hard to level thinset that is separated by gaps.