Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are (a) being atomic, i.e., not being decomposable as a union of two downward closed proper subsets or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are (1) words over a finite alphabet under the consecutive subword ordering, and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are characterizations of atomicity and well quasi-order for the subpath ordering on paths of a finite directed graph.