I'm talking primarily about sleeves used to repair scored cylinder walls, but feel free to mention aluminum engine cylinder sleeves as well.
What material are they made of and why? What's the wall thickness of the sleeve?

I suppose I might as well ask a somewhat related question: what material is used for bearing spacers (like for installing a 350 crank in a 400 block)? For that matter, how are they held in place?

 

To repair a cylinder, a cast iron sleeve is used. There are many many types, materials and thicknesses out there, wet sleeves, dry sleeves. Cast, steel, nikasil, etc.

Material usage is based off block parent material, desired wear characteristics, expansion rates, etc

Thickness depends on if it is a repair sleeve, or an OEM sleeve for an aluminum marine block, huge sleeves for industrial diesel apps.

Some are retained by a press fit, some are retained by o-rings and having the head sitting on them.

Research darton sleeve co, LA sleeve co, and Ohio sleeve co

The bearing spacers I assume are made out of mild steel, I never use them, I always have used aftermarket cranks with 350 mains or used special thick bearings. Having a spacer in there scares me.