So, in Skyrim, will we be able to fill stuff up in rivers and streams?
Like others posted, a real water simulation where a cup fills up with water if you drag it into the water is more or less impossible for a game (unless you have a preset animation for just a few items, but that's not really a water simulation). Won't happen. Activating water bodies so you can fill bottles in your inventory could easily be done though of course.
When we get out of water, will we be dripping wet? When it rains will we look like we are wet at all?
Could be in, could be done with a shader and some particle effects.
When the wind blows suring a storm, will our cloths and hair/fur move with the wind like the tree branches?
I don't think there will be cloth physics, wasn't this confirmed already? Pretty sure you'd have seen this in the trailer if it was in. Cloth physics are quite demanding, especially with the large amount of NPCs in Skyrim I doubt you will have anything like that. Probably like Oblivion.
Will snow catch in our hair/fur?
Could be done with a shader, but I imagine it would be pretty hard to do if you wanted it to look correctly and not just cover the whole body with white sprinkles. Probably possible, but I doubt it.