A hyperbolic 4-manifold has zero signature and hence is null-cobordant. However there exist hyperbolic 4-manifolds with trivial isometry group, and hence which cannot be a double of a 2-handlebody (such a double would admit an orientation-reversing isometry by Mostow’s theorem).

A hyperbolic 4-manifold has zero signature and hence is null-cobordant. However there exist hyperbolic 4-manifolds with trivial isometry group, and hence which cannot be a double of a 2-handlebody (such a double would admit an orientation-reversing isometry by Mostow’s theorem). This gives a partial answer to your first question (showing that there exists manifolds which are nullcobordant and not the double of a 2-handlebody).