These are indeed good questions, and while there is a very good corpus of answers to them, the analogy is not perfect.
0. The non-archimedean analogy
First of all, I would like to go back to the relative situation of a
surface $\mathcal X\to B$ fibered over a germ of curve $(B,b)$.
Then any local function $f$, resp. local section $s$ of a line bundle $\mathcal L$, may vanish along the special fiber with some multiplicity $m$, and this multiplicity is a valuation: the function $f\mapsto m=v(f)$
satisfies $v(f+g)\geq \min(v(f),v(g))$ and $v(fg)=v(f)+v(g)$.
The function $f\mapsto \exp(-v(f))$ behaves as a metric, except that it is non-archimedean.
Now forget the surface $\mathcal X$ and just remember about the generic
fiber, viewed as a curve over the complete valued field $F_b$, completion of the field of functions of $B$ with respect to the $b$-adic absolute value.
This furnishes a functor from pairs consisting of a surface $\mathcal X$ over $(B,b)$, line bundles on $\mathcal X$, to pairs consisting of a curve $X$ over $F_b$ and a metrized line bundle on $X$. This extends to vector bundles, in fact.
Under this functor, morphisms of vector bundles go to norm non-increasing morphisms of vector bundles.
This functor is “essentially” fully faithful (one needs some assumptions on $\mathcal X$, say it is normal). Up to blowing-up $\mathcal X$ along closed subschemes of the special fiber, it is “essentially” essentially surjective.
1. What is $H^0(\overline{\mathcal L})$ ?
The answer, which mimicks the above analogy, says that it is the
subset of $H^0(\mathcal L)$ of sections of norm $\leq 1$ everywhere.
It is a finite set with essentially no algebraic structure.
There are two propositions for its “dimension”.
The earliest one sets $h^0(\mathcal L)=\log \# (H^0(\overline{\mathcal L}))$. More recently, van der Geer, Schoof, Rössler, Bost, etc. have suggested to consider rather the theta-invariant:
$$ h^0_\theta(\mathcal L)= \log \left(\sum_{v\in H^0(\mathcal L)} \exp(-\| v\|^2) \right). $$
To answer one concern expressed in your statement of question 1. If you add $m_\sigma$ to the real component at $\sigma$, this multiplies the metric at $\sigma$ by $\exp(-m_\sigma)$. If $m_\sigma>0$, there will be more global sections, in good analogy with the fact that if you add an effective divisor to a divisor, the space of global sections increases.
2. Arithmetic degrees
In any case, rather than this specific $H^0(\overline{\mathcal L})$, it is rather convenient to remember the pair consisting of the free $\mathbf Z$-module $H^0(\mathcal L)$ and of its norm induced by the supremum norm of sections. (One can also introduce useful Euclidean norms by integrating local norms squared agains a fixed volume form; there are comparison results.)
This object is the analogue of a vector bundle over a curve, and can be given an arithmetic degree, satisfying good algebraic properties.
Minkowski's theorem is the analogue of the Riemann-(Roch) theorem giving non-zero sections in $H^0(\overline{\mathcal L})$ provided its arithmetic degree is large enough.
There is an analogue of Serre's duality theorem.
Actually, there are two analogues, according to your definition of $H^0$.
In the naïve one, it is given by an inequality (Gillet-Soulé, Israel J. Math.). In the theta-version, it comes from the Poisson summation formula and is an exact equality.
3. Arithmetic intersection theory
There I touch your second question. When $\mathcal X$ is a projective surface, there is a nice intersection pairing of line bundles. It can be defined geometrically, intersecting divisors. It can also be defined cohomologically, using Euler-Poincaré functions. (See, for example, the first chapter of Debarre's book on Higher dimensional geometry, for a rapid and clear treatment.)
The same can be done in Arakelov geometry. For arithmetic surfaces (the scope of your question), it is due to Arakelov, Faltings and Deligne; in general, this is due to Gillet-Soulé and Bismut-Gillet-Soulé. These authors define an arithmetic intersection of metrized line bundles, more generally of “arithmetic cycles”, and prove an arithmetic analogue of Grothendieck's Riemann-Roch theorem.
The difficulty with the mere statement of a Riemann-Roch theorem comes from the fact that the naive $H^0$, as we saw, are not algebraic objects, and that there is even no a priori definition of the higher cohomology groups. The idea is then to define the determinant of the cohomology, as a metrized line bundle on the base, whose arithmetic degree would be the analogue of the Euler-Poincaré characteristic.
4. Applications
I cannot resisting mentioning a bunch of applications.
a) This theory fits extremly well, and makes precise, Weil's theory of heights. In Weil's theory, height functions are defined up to bounded ambiguities; here, one has actual functions under the hand.
b) Gillet-Soulé proved an analogue of the Hilbert-Samuel theorem, giving an estimation for the cardinality of $H^0(\overline{\mathcal L}^n)$, where $n\to\infty$. This theorem has been extremly useful, for example in the proof of Bogomolov's conjecture.
c) Zhang proved an analogue of the Nakai-Moishezon theorems, that is, criteria on $\overline{\mathcal L}$ implying that large powers will be generated by sections of norm $<1$. (Basically, the height of every closed subvariety has to be $>0$.)
d) The precise formulas coming out of the arithmetic Riemann-Roch theorem furnish beautiful arithmetic formulas relating heights and special values of $L$-functions. (See the work of Maillot-Rössler, for example.)
e) Bost proved analogues of Lefschetz's hyperplane section theorem on Riemann surfaces, implying for example the triviality of the fundamental groups of some arithmetic surfaces.
f) Bost and myself proved generalizations of the Borel-Dwork criterion, viewed as analogues of the Hironaka-Matsumura theory (algebraization of formal schemes).
g) The formalism is also very useful to formulate in a geometric way the proofs of transcendental number theory (Bost's slope method). It has been used, for example, by Bost-David (explicit version of Masser-Wüstholz isogeny estimates), Bost (generalization of Chudnovsky/André's results about Grothendieck's conjecture to foliations), Gasbarri and Herblot (Schneider-Lang type theorems), etc.
It doesn't so much represent a dimension of a cohomology group as it does an Euler characteristic.
More precisely, it's based on Grothendieck's generalization of the Riemann-Roch theorem to families. Given a proper map $\pi: Y\to X$ and a line bundle $L$ on $Y$, we could of course expect Riemann-Roch or a generalize to give us a formula for the Euler characteristic of the fiber $\sum_i (-1)^i \dim H^i(Y_x, L)$, which can be calculated using the cohomology sheaves $R^i \pi_* L$. But in fact this is forgetting a lot of information, and its better to calculate the class $\sum_i (-1)^i [ R^i \pi_* L]$ in $K$-theory, which contains this dimensional information, but also other information.
In the case when the base $X$ is a smooth curve, the $K$-theory group is $\mathbb Z \times \operatorname{Pic} X$, where the $\mathbb Z$ comes from calculating the rank of the coherent sheaves $R^i \pi_* L$, and the $\operatorname{Pic} X$ comes from the determinant.
So determinant-of-cohomology is a generalization of the K-theoretic information you get by taking cohomology down to the base curve.
Moreover, one piece of information that is contained in the K-theory classes of the cohomology groups $R^i \pi_* L$ is their Euler characteristics $\chi(X,R^i \pi_* L)$, and we have $\chi(Y,L) = \sum_i (-1)^i \chi(X, R^i \pi_* L)$. So in fact, in the geometric setting of a family of varieties of a curve, the $K$-theory class determines the Euler characteristic of the absolute cohomology groups - it's something like the rank times $1-g$ plus the degree of the determinant-of-cohomology.
The rank is just the usual Euler characteristic of the generic fiber, so it's not so arithmetically interesting, so we can think of the determinant of cohomology as a substitute for the Euler characteristic of the missing arithmetic cohomology theory.
Best Answer
The Green's function is used not to measure distances in the surface but to measure distances in the line bundle. A Green's function on $X_{\mathbb C}$ that blows up at $D$ can be used to measure sections of $\mathcal O(D)$. Indeed, if we represent a section of $\mathcal O(D)$ as a holomorphic function $f$ on $X_{\mathbb C}$ with poles at the points of $D$, and $g$ is a Green's function on $X_{\mathbb C}$ with poles at $D$, then $|f(x)|^2 e^{- |g(x)|}$ is a smooth nonnegative function that vanishes only where $f$ vanishes as a section of $\mathcal O(D)$ - i.e. where it lies in the image of the natural map $\mathcal O(D-P) \to \mathcal O(D)$. In fact it is easy to see that this is a Hermitian form on the space of sections. So the Green's function gives a metric on the line bundle.
We need a metric at $\infty$ in an Arakelov line bundle because we already have a metric (in the form of a $p$-adic valuation) at every other place - any section of $\mathcal O(D)$ on $X_{\mathbb Q_p}$ has a well-defined $p$-adic valuation at each point of $x\in X_{\mathbb Q_p}$, where a local generator of the line bundle at the reduction mod $p$ $\overline{x}\in X_{\mathbb F_p}$ is given absolute value $0$ - because such a generator is also a generator of the fiber of $\mathcal O(D)$ at $x$, this determines the valuation on every section.