I think both questions are open. The somewhat sad state of affairs is that there are nontrivial TOP 4d s-cobordisms that are either nonsmoothable or not known to be smoothable, and there are smooth 4d s-cobordisms
that may well be products. No h-cobordisms with nontrivial torsion seems to be known.

It seems the state of the art is described in the introduction to a paper by Weimin Chen "Smooth s-cobordisms of elliptic 3-manifolds" , JDG (2006), where references can be found.

Convention: all cobordisms below are of dimension 4 (i.e. have 3-manifold boundaries).

There are only finitely many orientable TOP s-cobordisms
with the boundary the same elliptic 3-manifold and in some cases there is a complete classification (Cappell-Shaneson, Kwasik-Schultz).

There are infinitely many non-orientable TOP
s-cobordisms (Matsumoto-Siebenmann, Kwasik).

Kwasik gave (modulo now known elliptization conjecture)
a list of finite groups such that any 4-dimensional topological h-cobordism with the fundamental group on the list must have trivial Whitehead torsion, see
"On four-dimensional h-cobordism".
Of course, the Whitehead group itself of those finite groups is often nontrivial.

Cappell-Shaneson constructed examples of smooth s-cobordsims with ellipltic 3-manifold boundaries, but it is unknown whether the cobordisms aren't products, and partial results of Akbulut indicate they are probably smooth products.

Chen proved that a symplectic s-cobordism with elliptic boundaries is a product, and conjectured that a smooth s-cobordism is a product if and only if its universal cover is a product.