If you are new to wood processing, I would recommend a fixed blade knife and a saw. The saw will crosscut and the knife will split by batoning the wood. The size of the knife and the size of the saw will depend on the size of wood you are processing. My choice of saw would be the trailblazer sawvivor. If you want something smaller, a bacho laplander is a good choice.
With an axe, there isn't really a need for a saw since it can crosscut wood well. Axes are best for chopping up long pieces of wood for a big, long log fire. If you are going to tarp camp, you shoulder consider a big ground fire. A stove isn't going to do much unless it's in a enclosed tent. But then you have the issue of sparks flying around and I would advise you to have a cheap bivy and a cheap-ish tarp if you are going to sleep next to a fire.
The issue isn't just the wood being under snow. Any wood that isn't vertical standing will get wet from rain and freeze. You will still be able to find dead branches on trees for kindling. I'm not sure if they would be dry or not since I haven't snow camped before.
Keep in mind that cut wood looks kinda ugly and artificial until it falls over and rots, so I wouldn't do that in a popular campsite or along a trail.