Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used the compactness theorem to verify that nonstandard models of PA exist, we can appeal to the Löwenheim–Skolem theorem to ensure that we have countable ones. We can then verify that these models form unbounded dense linear orders of $\mathbb{Z}$ blocks beyond the standard portion and use a back and forth argument to confirm that the nonstandard parts must always be order isomorphic.

My questions concern the structure of uncountable nonstandard models of PA. The Löwenheim–Skolem theorem can be used to show that we have nonstandard models of arbitrarily large cardinality so:

(1) Is there an example of a set $M$ of the form $\mathbb{N} + \mathbb{Z} \cdot D$ where $D$ is a dense linear order such that $M$ is *not* a model of PA?

(2) Can we find arbitrarily large $\kappa$ and nonstandard models of PA of size $\kappa$ such that there will be ~~a~~ **an unbounded** well-orderable subset of size $\kappa$ respecting the ordering relation of the nonstandard model (i.e. an order isomorphism between an ordinal and a subset of the model)?

(3) If the answer to (2) is no, what types of related results can we get?