How does the length of the studs on winter tires affect traction on ice?
They're not really that short; compare them to the studs in studded car tires, for example.  A tiny bit of metal gives you a huge amount of traction on ice.  The real trick is that, unless you're purely riding on frozen lakebeds, you need studs small enough to deform into the tire when riding on pavement.

In icy conditions I ride 37mm Nokian W240's, they fit just fine under my 65mm Cascadia 29er fenders.  In snowy conditions, however, the snow can build up under the fender and make it difficult to ride.

If you are planning on riding long distances in deep snow there are only a small number of options, chief among which are the the Surly Endomorph or Larry tires. At 3.5" wide and at low PSI there is no comparing with a smaller tire.  Of course, you need a special bike and rims to run those beasts.