I would go screws. Why risk the staples ever coming loose? Before the new layer goes down make sure the squeeks in the floor are taken care of.

I don't think you need to fill them. However, should you want to do that anyway, then just get some cementious filler, such Armstrong Feather Finish or an equivalent product. Mix with water as directed, apply and then work it flush to the surface. These products dry hard and fast. More important is to make sure your substrate is flat to within the flooring manufacturer's guideline.

I do not believe AC plywood is suitable for anything vinyl as it is not a solid core plywood. It used to be vinyl had to be over luan plywood although I am sure that has changed today but I still do not think AC plywood is acceptable.

What is the nature of the failure on the plywood you're referring to? I've probably laid hundreds of thousands of square feet of plywood over concrete and can't think of one failure I could attribute to the installation method (i.e. the failures have all been caused by other factors, like a flood). Granted we go a little overboard in that we flat trowel cold bond adhesive on the concrete at 50'/gallon, lay in heavy poly sheeting, trowel another layer of mastic over that at 32'/gallon, lay in the plywood, shoot it down with 1-1/2" hardened coil nails and fasten any loose areas with tapcons. Before sheet vinyl fell out of favor we used it under plywood all the time in conjunction with adhesive. I've had a few jobs recently where the client wanted something better that visqueen so we used a membrane like Traxxshield99 with seam tape. The plywood subfloors I have seen fail due to installation method, are almost always laid dry over a single layer of black felt paper (which is not a suitable moisture barrier), then shot or screwed down. I don't think there's any need to build up a subfloor like that and affect the relationship to doors and steps, but if the advantages were deemed to be beneficial, I'd recommend a moisture barrier, 1x4s fastened to the concrete 12" on center and a layer of 3/4" plywood over that, with the joints laid on the 1x4s. This can be a little complicated because it would be nice to have p.t. lumber for the 1x4s but they're generally soaking wet upon purchase and there may be an issue using steel fasteners.