These type of questions have been studied in detail by P. Nyikos, for example in his article Non-metrizable manifolds in the Handbook of Set Theoretic Topology, where he shows that there are $2^{\aleph_1}$ non-homeomorphic ``long planes''“long planes”, that is, simply connected $\omega$-bounded surfaces. (He actually shows that there is such a family of non-homeomorphic longpipes, but it is easy to sew a disc in a longpipe to obtain a longplane. See the article for definitions.) Hence, as noted in a previous answeranswer by new account, there are many non-homeomorphic surfaces that are maximal in OP's sense.
Nyikos also showed (Thm 6.1) in
On first countable, countably compact spaces III: The problem of obtaining separable noncompact examples, Open Problems in Topology, J. van Mill and G.M. Reed (Editors), 1990, pp. 130-161130–161.
that a $n$-manifold $M$ is properly contained in another $n$-manifold iff there is a closed copy in $M$ of the closed unit $n$-ball with a deleted point in its boundary. As a corollary (Cor. 6.2), a pseudocompact manifold is maximal. (Recall that a space is pseudocompact iff any real valued map on it is bounded.) Nyikos' Example 6.7 in the same paper is a maximal surface whose properties depend on the model of set theory: under the continuum hypothesis, it can be made countably compact, and under $\mathfrak{p}>\omega_1$, it cannot be pseudocompact. Hence, the relationship between pseudocompactness and maximality can sometimes be a subtle matter.
I don't know the answer to Question 2 but I strongly suspect that it is negative.
The answer to question 3 is clearly no, as also noted in the same previous answeranswer by new account, but a small variation of it seems more interesting:
Question 3': If $N$ is a Jordan surface (in OP's sense) properly contained in two maximal Jordan surfaces $M_1,M_2$$M_1$, $M_2$ and dense in them, are $M_1$ and $M_2$ homeorphic ?
The answer to this question is also negative: Nyikos' example 6.7 is the union of a copy of the longray and the $2$-sphere with a deleted point. The long ray is in a sense ``pushed inside''“pushed inside” the hole left by the deleted point. One can delete more points and push copies of the longray in each scar. This gives non-homeomorphic maximal surfaces which all have a dense subset homeomorphic to open $2$-disk. (There are certainly simpler counter-examples to Question 3'.)
As to Question 4, I admit that I do not really understand your definition of Prüfer Surface (maybe because I am on holydaysholidays and somewhat slow at thinking), but I guess that it must be the classical example of attaching uncountably many copies of $\mathbb{R}$ to the half plane, yielding a surface whose boundary is these uncountable copies of $\mathbb{R}$. If that is the case, I believe that another example in the same paper by Nyikos (ex 6.3) can be adapted to obtain a pseudocompact (hence maximal) surface containing the Prüfer surface, however the construction works only when $\mathfrak{b}=\omega_1$. (Nyikos' example is described in more details in D. Gauld's book ``Non“Non-Metrisable Manifolds''Manifolds”, ex. 1.29 on page 15).
Note: $\mathfrak{b}$ and $\mathfrak{p}$ are classical small uncountable cardinals defined for instance in Nyikos' above cited article.
