Question: The class $W$ of well-ordered sets is not first-order axiomatizable. However, it does have an associated first-order theory $Th(W)$. I define a pseudo-well-ordered set to be an ordered set $(S;\leq)$ that satisfies $Th(W)$. My question is, is the class of pseudo-well-ordered set closed under taking substructures? If so, I want a proof. If not, I would like an explicit pseudo-well-ordered set $(S;\leq)$ and a suborder $(T;\leq)$ of $(S;\leq)$ such that $(T;\leq)$ is not pseudo-well-ordered.

Answer: 
$\omega+\zeta$ satisfies all the first-order sentences satisfied by $\omega$, but $\zeta$ does not satisfy "there is a smallest element".
Take any countable non-standard model of $\sf PA$ which is elementarily equivalent to the standard one, it is known that the order type of such model has the form $\Bbb{N+Q\times Z}$, therefore it contains a suborder isomorphic to $\Bbb Q$, which is clearly not satisfying the theory of pseudo-well-orders.