Concerning question (1).
I became very interested in this question last year---obsessed
with it, actually---when I found myself unable to prove that any
of the natural-seeming examples were actually instances of
incomparability (for example, none of the approaches suggested in the various comments actually work). After my numerous attacks on it failed, I began
seriously to doubt the strong intuition underlying the question,
that there should be incomparable models. Eventually, I was a able to show that indeed, any two countable models are comparable by embeddability. My paper is available at:
The main theorems are:
Theorem 1. Every countable model of set theory $\langle
M,{\in^M}\rangle$ is isomorphic to a submodel of its own
constructible universe $\langle L^M,{\in^M}\rangle$. Thus, there
is an embedding
$$j:\langle M,{\in^M}\rangle\to \langle L^M,{\in^M}\rangle$$
that is elementary for quantifier-free assertions in the language
of set theory.
The proof uses universal digraph combinatorics, including an
acyclic version of the countable random digraph, which I call the
countable random $\mathbb{Q}$-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading eventually to what
I call the hypnagogic digraph, a set-homogeneous, class-universal,
surreal-numbers-graded acyclic class digraph, which is closely
connected with the surreal numbers. The proof shows that $\langle
L^M,{\in^M}\rangle$ contains a submodel that is a universal
acyclic digraph of rank $\text{Ord}^M$, and so in fact this model
is universal for all countable acyclic binary relations of this
rank. When $M$ is ill-founded, this includes all acyclic binary
relations.
The method of proof also establishes the following, which answers
question (1). Version 2 on the archive, which will become visible
in a few days, cites this question and Ewan Delanoy.
Theorem 2. The countable models of set theory are linearly
pre-ordered by embeddability: for any two countable models of set
theory $\langle M,{\in^M}\rangle$ and $\langle N,{\in^N}\rangle$,
either $M$ is isomorphic to a submodel of $N$ or conversely.
Indeed, the countable models of set theory are pre-well-ordered by
embeddability in order type exactly $\omega_1+1$.
The proof shows that the embedability relation on the models of
set theory conforms with their ordinal heights, in that any two
models with the same ordinals are bi-embeddable; any shorter model
embeds into any taller model; and the ill-founded models are all
bi-embeddable and universal.
The proof method arises most easily in finite set theory, showing
that the nonstandard hereditarily finite sets $\text{HF}^M$ coded
in any nonstandard model $M$ of PA or even of $I\Delta_0$ are
similarly universal for all acyclic binary relations. This
strengthens a classical theorem of Ressayre, while simplifying the
proof, replacing a partial saturation and resplendency argument
with a soft appeal to graph universality.
Theorem 3. If $M$ is any nonstandard model of PA, then
every countable model of set theory is isomorphic to a submodel of
the hereditarily finite sets $\langle \text{HF}^M,{\in^M}\rangle$
of $M$. Indeed, $\langle\text{HF}^M,{\in^M}\rangle$ is universal
for all countable acyclic binary relations.
In particular, every countable model of ZFC and even of ZFC plus
large cardinals arises as a submodel of
$\langle\text{HF}^M,{\in^M}\rangle$. Thus, inside any nonstandard
model of finite set theory, we may cast out some of the finite
sets and thereby arrive at a copy of any desired model of infinite
set theory, having infinite sets, uncountable sets or even large
cardinals of whatever type we like.
The article closes with a number of questions, which you may find
on my blog post about the article. I plan to make
some mathoverflow questions about them in the near future.