Is there a technique as such? - I'm assuming you don't just lash them on and plod off?
No, more or less you do just lash them on and plod off
If you're thinking of something hard like putting rollerblades on it's nothing like that. I might have some familiarity in terms of setting a track or moving on steep ground but it's all simple stuff.
How deep is the snow under you? - How far do you sink into it with the snowshoes on?
I've been out in snowpacks up-to 4 meters deep, how far you sink depends on the snow, rarely beyond boot tops. The snowshoes make it easy to move along though.
Can the snow be so soft that the snowshoes don't work, and just end up meaning that every lift of your foot comes with a weight of snow on top of it?
You can get a lot of soft snow on the surface but it doesn't build up on the snowshoe. Sometimes in spring there's some complex layering in the snow and you can crash through a couple of layers. These conditions are dangerous for other reasons though so you need to be skilled or with someone who knows what they're doing.
This is an MSR Denali EVO Ascent snowshow, it's a fairly top-end model and you strap a standard boot into it. I happen to use a more mountaineering orientated boot as I'm out a lot and I need something fairly stiff if I want to move in mixed ground or kick steps etc. Any standard walking boot would be just fine.
You can see on the underside the MSR has steel rails with sabretooths for grip on ice and lateral struts which give a lateral strength and work as a track. Under the toe I've a crampon that's pivoting under the ball of the foot, I can basically front point up something using those spikes. The MSR is a high end model and it's laterally fairly stiff so if I traverse it doesn't flex laterally too much.
It's hard to say how steep you can move on, it depends on the snow and terrain. On steep ground we'd zig-zag up kick-turning to alter direction. Moving down on short sections it's easier to "bound" down like Amy is doing in the third and fourth photographs above.
Another neat feature of the Denali models is you can add float tails as in the picture. How much you sink in snow is a function of surface area so if you increase the deck area you'll float in deeper, fresher snow a lot more.
This is simple and fun, winter walking in the alps would be out the question without some enhanced flotation. There's also more to do in the alps in winter than buying an expensive lift pass and skiing, in fact even a pair of snowshoes like mine although relatively expensive is still cheaper than many week long ski lift passes.
All that said, winter in the alps is a serious environment, in deep winter your chances of surviving a night out aren't great and there's a huge risk from avalanches once you're out of secured areas. Both Julie and I are wearing avalanche transceiver in those photo's and the rucksacks have probes and shovels in them. Anyone could rent a pair of snowshoes and have a go around some of the prepared trails though, it's just if you're moving away from them you need to be aware of the risks or join a guide or leader.