Minecraft – How to prevent snow steps from Snow Golems

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.

Might as well provide an answer based on the source code here.

A snow golem is an EntitySnowman, which inherits its logic from EntityGolem. It will never despawn. (That is, calling canDespawn() on an EntitySnowman always returns false.) So, your snow golem is absolutely not despawning on its own.

Why is it disappearing? That I don't know. If it gets touched by fire, water, rain, or attacked by any other mob, it can easily die. If none of those things are happening, then it probably is some kind of suffocation glitch - maybe it wanders into a block and gets stuck somehow.