Can snow fall on half-steps?

It's not possible for snow to exist on slabs/half steps. A slab is only ½ blocks high, but as far as the game is concerned, it inhabits an entire block. The same goes for the snow layer, which is also a block — except other blocks you place can go through and replace the snow layer block. So snow on a slab would have to float half a block above ground.

Try placing a cobblestone block on a slab, same thing:

Double slabs are a different block entirely. Their height is exactly one block, so snow can exist/will fall on it.

Edit to add:

Are there any solid blocks that snow cannot stick to?

Those would be non-opaque blocks: Glass and ice. In Fancy Graphics mode, snow will also not fall on leaves.

I just tested this with a snow golem in 1.5.2 (right before the horse update)

Snow was placed on glass, glowstone, gravel, sand, and end stone just fine.

However, upside down stone slab and upside down nether brick slab did not work, even though their top is flush with a normal full sized block.

You can hack snow in, and it will 'float' above halfsteps.

It will not fall naturally.