I've got Eagle GT's on my ZX3, and now the rears have succumbed to the infamous Focus cupping problem. I've got the suspension problems fixed and so the next thing to buy are two new tires. I've never been thrilled with Eagle GT snow handling, and am willing to try a set of truly all-season tires (maybe Continental DWS), with the understanding that I won't see their full benefit since I'm only putting them on half of the car.

Tire stores, of course, want to upsell every customer with four new tires. A couple places used scare tactics, telling me "Don't mix performance and all-season tires!", saying that it's gonna wreck the car's handling with Eagle's on the front and all-season's in back (or vice versa), implying that it's gonna cause an accident.

Is there any merit to that? As far as dry conditions go, I know they're full of BS. But, I'm concerned about how it might affect stopping and steering on rain, snow, and ice.

stopping in the snow with only 2 snow tires on the front tires may cause some stability issues and the car might swap ends.
in the dry, not so much, but it still might at the limits of traction (evasive manouvers).

IMO, i would try keep all 4 tires similar. i dont think they need to be same brand. but i personally wouldnt mix summer/allseason/winter in the winter, although i might mix summer/allseason in summer if i was in a bind.

Mixing tires can affect handling. It's primarily due to sidewall flex. Some tires are just stiffer than others. Mixing them can result in a looser front or rear end. The Conti's are known to have somewhat softer sidewalls than most other high performance all season tires (possibly their only flaw), but this softer sidewall also is partially responsible for their good performance in poor conditions. I remember when I was young (and dumb) I threw some new fangled radials on the rear end of my 70 Barracuda. It still had some Goodyear wide oval (biased ply) tires up front. The car was almost unmanageable over 40mph. It was scary with the rear end was swaying all over the place. This is the extreme example of mixing tires. On the other hand, I had a blow out in my Miata some years ago and I didn't want to drive on the donut spare and the blown tire was unfixable. I ended up gettingt a cheap replacement tire as the other three were in very good condition. That was a mistake as well as it changed the cars handling quite a bit whether it was on the front or rear. My advise, it's not worth doing unless you absolutely have no other choice.

And by the way, if your rear tires are truly "cupping" (a scalloped appearance on the edge of the tire) you have a tire balance or worn shock absorber problem (usually the latter). Alignment issues generally result in an evenly worn and smooth tire shoulder.