I've heard of guys having issues with that because of the snow not being able to pass through as the jaws close, so they find a snapped trap with a big compacted snowball in the trap because the snow displaced the paw as it compressed. I've never personally tried it, I use wax paper.
Exactly, I've heard that in the very cold temps they get some places that plastic just shatters, but myself and everyone I know who has tried it here has had the exact problems you specified.
Put wax paper under your trap when you set it in the snow, depending on the snow conditions you may have to compact the snow to set the trap, or you may just be able to shove it down and wiggle it a bit to get it solid. Then cover with wax paper and either use a shovel/scoop etc. to scoop up some snow with as little disturbance as possible and slide it off the scoop over the trap, or my preferred method when possible, just put a couple small dabs of snow in a couple corners of the wax paper to hold it down long enough for it to snow over. Obviously to do that you need to be setting either in the snow, or just before it is going to snow. That is the best way when possible, disturbed snow always freezes, but if you can let it snow over naturally it will be as soft as all the undisturbed snow around, and the fresh snowfall will blend all the disturbance you made while making the set.
Assuming you have similar weather to what I do here in Northern Idaho, which will be similar to what Tramp has in SE Alaska. Snow sets are quick and simple to put in, just throw a drag and chain out in the snow and it sinks out of sight, no trap beds to dig, most of the time you can just lay down a chunk of wax paper, shove the trap down on top of it and wiggle it a bit to have it solid (sometimes with crappy snow you have to scoop out a hollow under the center of the trap to get it solid and not rocking) cover it with wax paper and either snow or simply let Ma Nature cover it. ...But they are constant, high maintenance to keep them working, they melt out, they snow two feet over them, or it rains on them and then freezes solid. Or all three.
You can really slam in a bunch of sets in a day in the snow, but you may spend more time a check or two down the road redoing them all than you did putting them in the first time. If possible, best to have a variety, with some sets under tree cover and some in the open, it has been my experience that those under tree cover will get dripped on and froze up faster when it warms up than those in the open, but those in the open will be out of commission much faster when it dumps snow. At least with a variety you may have some sets that are working.