Another example, like JSE's, that comes already equipped with a Belyi map but is not as familiar as modular curves and Fermat curves: For any relatively prime integers $m,n$ with $0<m<n$, and any subgroup $G$ of $S_n$, the curve that parametrizes trinomials $x^n + a x^m + b$ up to scaling with Galois group contained in $G$. The Belyi map is the invariant $a^n/b^{n-m}$ of the trinomial, and its degree is $d=[S_n:G]$; it is branched at $0$, $\infty$, and $(-n)^n/(m^m (n-m)^{n-m})$. One may assume $m \leq n/2$ (by symmetry with respect to $x \leftrightarrow 1/x$, $m \leftrightarrow n-m$). Some nontrivial examples with $n=5,7,8$ are given explicitly at http://www.math.harvard.edu/~elkies/trinomial.html; the subsequent paper with N.Bruin on the cases $(m,n) = (1,7)$ and $(1,8)$ with $d = 30$ is
Nils Bruin and Noam D. Elkies, Trinomials $ax^7+bx+c$ and $ax^8+bx+c$ with Galois Groups of Order 168 and $8 \cdot 168$, Lecture Notes in Computer Science 2369 (proceedings of ANTS-5, 2002; C.Fieker and D.R.Kohel, eds.), 172-188.
(These examples all have $G$ transitive, but the construction works for all subgroups $G$.)
Here are some observations. I include the case $g=1$ (even if $X$ has no rational point).
Denote by $\hat{\mathcal X}$ the (proper) minimal regular model of $X$ over the $O_K$ and let $\mathcal X$ be the smooth locus of $\hat{\mathcal X}$.
(1) If the Néron model exists, it is equal to the smooth locus $\mathcal X$ of the minimal regular model.
(2) If the fibers of $\mathcal X$ over $O_K$ have no rational irreducible component (e.g. if $X$ has good reduction), then $\mathcal X$ is the Néron model of $X$.
(3) (localization) If Néron models exist over DVRs, then they exist over any Dedekind domain.
(4) (base change) Let $R$ be a DVR. Let $R'/R$ be an extension of DVR such that an uniformizing element of $R$ is also an uniformizing element of $R'$ and such that the residue extension is separable (e.g. $R'$ can be the completion of a strict henselization of $R$). If the Néron model exists over $R'$, then the Néron model exists over $R$.
(5) You were right to not include the case $g=0$. The projective line doesn't have Néron model.
(6) Let $Y$ be a smooth scheme over a noetherian regular scheme $S$, let $Z$ be a regular scheme, flat and of finite type over $S$ and let $f: Y\to Z$ be a morphism. Then $f(Y)$ is contained in the smooth locus of $Z/S$. In particular, the canonical map ${\mathcal X}'(O_K)\to X(K)$ is bijective if $\mathcal X'$ is the smooth locus of (any) proper regular model of $X$.
(7) If $g=1$, then $\mathcal X$ is the Néron model of $X$.
Proof. Sorry I can't give all details by lack of energy and because it would be pretty unreadable in MO.
(1) Let $\mathcal N$ be the Néron model. Embedd it in a proper flat model, solve its singularity without touching to the regular locus (which contains $\mathcal N$). Then we get a proper regular model $\hat{\mathcal N}$ containing $\mathcal N$ as an open subset. The identity on $X$ extends to morphism $\hat{\mathcal N}\to \hat{\mathcal X}$. By (6), this morphism induces a morphism $\mathcal N\to \mathcal X$.
Then $\mathcal X$ satisfies the universal Néron mapping property. By the uniqueness of Néron model, we get $\mathcal N\simeq \mathcal X$.
(2) Let $\mathcal Y -\to \mathcal X$ be a rational map defined over $K$ with $\mathcal Y$ smooth (regular is enough). The projection $p: \Gamma\to \mathcal Y$ is birational. Let $y\in Y$ and suppose $\Gamma_p$ is not finite. By a theorem of Abhyankar (the base scheme is excellent here, otherwise, localize and pass to the completion and use (4)), the components $E$ of $\Gamma_p$ are uniruled. But $E\to \mathcal X$ is a closed immersion, so $E$ is a rational curve in a close fiber of $\mathcal X$. Contradiction. Thus $p$ is quasi-finite biratonal and surjective. As $\mathcal Y$ is normal, $p$ is an isomorphism by Zariski's Main Theorem and the rational map we consider is actually defined everywhere. So $\mathcal X$ is the Néron model.
(3) The curve $X$ has good reduction away from finite many places. Using (2) for good reduction places and by gluing with Néron models over bad reduction places, we get a global Néron model over $O_K$.
(4) First the formation of the minimal regular model (and its smooth locus) is compatible with such base change. So if $\mathcal X\otimes R'$ satisfies the universal Néron mapping property over $R'$, then so does $\mathcal X$ over $R$ by faithfully flat descent for the definition domain of rational maps.
(5) Fix a model $\mathbb P^1_{O_K}$ of $\mathbb P^1_K$. They are plenty of endomorphisms of the generic fiber which don't extend to $\mathbb P^1_{O_K}$ (e.g. $[x,y]\mapsto [x, py]$). This shows that $\mathbb P^1_K$ has no proper smooth Néron model. The general case can be proved similarly with some extra works.
(6) Sketch: Consider $Y\times_S Z\to Y$. It is enough to show that its sections have images in the smooth locus (over $Y$), then use descent of smoothness (easy). The left hand side is regular because it is smooth over the regular scheme $Z$, and the right hand side si regular because it is smooth over the regular scheme $S$. So we can reduced to the case of flat morphism of finite type $W\to Y$ between two regular schemes. Let $y\in Y$ and let $w\in W$ be its image by a section $Y\to W$. Then
$ O_{W,w}\to O_{Y,y}$ is a surjective map of regular local rings. Its kernel is generated by {$t_1, \dots, t_d$}, a part of a system of coordinates of $O_{W,w}$. So the maximal ideal $m_w$ of $O_{W,w}$ is generated by $t_1, \dots, t_d$ and $m_y$. Thus the maximal ideal of $O_{W_y, w}$ is generated by the images of $t_1, \dots, t_d$. The flatness of $W\to Y$ implies that $W_y$ has dimension $d$ at $w$. So $W_y$ is regular at $w$. It is in fact smooth because $w$ is a rational point of $W_y/k(y)$.
Another proof is to use $\Omega^1_{W/Y}$ and the fact that the image of a section is locally complete intersection.
Application: if $\hat{\mathcal X'}$ is a proper regular model over $O_K$, by the valuative criterion, $\hat{\mathcal X'}\to X(K)$ is bijective. But we just saw that the LHS is $\mathcal X'(O_K)$.
(7) We can work over a DVR $R$. If there exists a smooth $R$-scheme $\mathcal Y$ with non-empty special fiber and a morphism $\mathcal Y_K\to X$, then $\mathcal Y_K$ has a point in an étale extension of $R$. So $X$ has
a point in an étale extension. By (4), we can thus suppose $X(K)\ne\emptyset$. So it is an elliptic curve, and Néron showed that $\mathcal X$ is the Néron model. If such $\mathcal Y$ doesn't exists, then $\mathcal X$ trivially satisfies the Néron mapping property.
Best Answer
Dear Ariyan, the elliptic curve with equation $$y^2=x^3+6$$ has Faltings height $$-(3/2)\log(\Gamma(1/3)/\Gamma(2/3))+(1/4)\log(3)=-0.748752...;$$ the curve of genus $2$ with equation $$y^2+y=x^5$$ has Faltings height $$ h_{\rm Fal}(C_{\bar{\bf Q}})=2\log(2\pi)- {1\over 2}\log\big(\Gamma(1/5)^5\Gamma(2/5)^3\Gamma(3/5)\Gamma(4/5)^{-1}\big) $$ $$ \approx -1.452509239645644650317707042; $$ For the first example, see Deligne, "Preuve des conjectures de Tate et Shafarevich", Séminaire Bourbaki. For the second one, see Bost, Mestre, Moret-Bailly, "Sur le calcul explicite des 'classes de Chern' des surfaces arithmétiques de genre $2$", Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 69–105.
Another explicit formula that should allow you to produce elliptic curves of arbitrarily large Faltings height is the inequality $$ |h(j_E)-12h_{\rm Fal}(E)|\leqslant 6\log(1+h(j_E))+47.15 $$ See paragraph 5. of the article "Serre's uniformity..." by Bilu and Parent for references.
Something else you can do is make numerical experiments with formula in Conj. 3 of the article of Colmez, "Hauteur de Faltings..." (Compositio), which is true (without $\log(2)$ factor !, see A. Obus, arXiv:1107.0684) if the CM field is abelian over $\bf Q$. In that case, the Artin $L$-functions become Dirichlet $L$-functions and can be computed explicitly in terms of values of the Gamma function using the Hurwitz formula.
This is not a complete answer but I hope that it helps.