Hmm... ice patches: relax, don't brake, don't steer, don't pedal, don't tense up and you'll probably stay on. Oddly, some ice, in the Peak for example, where there's a lot of grit mixed in, is deceptively grippy.
Snow is just super variable depending on the snow and how wet/deep/soft/compressed/frozen/not frozen it is. At its best it's improbably grippy, at worst, completely unrideable porridge.
Some wet snow will clog SPDs - flats can work better, but it depends on the snow really. I've also carried and used micro-crampons in proper iced-up conditions before now.
If there's a proper ice/snow cold snap, studded tyres work well and you can ride on sheet ice as long as you're smooth, also good on hard-packed snow, but draggy and noisy on roads. Expensive too, but let you ride on days when you'd otherwise be stuffed. A few winters back when the trails round here were snowed up for a couple of months, I rode a lot of backroads and lanes - Peak District - on a hardtail with ice tyres on and it was surprisingly good fun.
Also, a little light goes a long way in snow, which is nice, but a little eerie.
Roads? Studded tyres on a cross or mountain bike when things go ballistic, rest of the time, watch out for patches of ice in shaded areas, dips etc. If it's cold enough for it to form and hang around be prepared to hop off and walk/slide past it would be my take. Or resign yourself to the turbo...
Overall I think you just have to play it by ear. Ride smoothly. Enjoy