A combinatorial motivation is the n! conjecture, whose proof by Haiman uses Hilbert schemes. An account of this work written by Haiman for the Current Developments in Mathematics conference in 2002 is at math.berkeley.edu/~mhaiman/ftp/cdm/cdm.pdf. Haiman emphasizes at the start of the paper that the main geometric results which had to be proved were motivated by combinatorial evidence. Around the time that Haiman first announced his results on the n! conjecture (before he moved to Berkeley) I had heard from other people that this conjecture motivated Haiman to learn modern algebraic geometry. Haiman's response to receiving the Moore prize in the AMS Notices April 2004, p. 432, more or less seems to confirm this, so it's analogous to the way that the Weil conjectures were a concrete open problem which motivated Grothendieck's work.
Your answer is correct, except that in the example you gave, one has to adjoin far more numbers than you described in order to get to that ordered Pythagorean not-Euclidean field K.
That example is described on p.594 of the fourth edition of my book Euclidean and Non-Euclidean Geometries: Development and History (W.H. Freeman, 2008). There I called such a plane where parallel lines have a unique common perpendicular an HE-plane, abbreviating "halb elliptisch" from Pejas' classification article. Since you are only interested in the Archimedean case in order to get a natural metric once a unit segment is chosen, your HE-plane satisfies the acute angle hypothesis (by the Saccheri-Legendre theorem). You then describe an HE-plane accurately as the interior of a virtual circle in the affine plane over an ordered Archimedean Pythagorean not-Euclidean field K. An Archimedean ordered field is a subfield of R (up to isomorphism), so when you metrically complete it, you get R.
To make your argument completely rigorous, you would have to prove that the metric completion of an Archimedean H-plane is again an H-plane. Then, since it is complete, it must be either the real Euclidean or the real hyperbolic plane (in the HE-plane case it is the real hyperbolic). As I pointed out on p.594 of my book and as you indicated, if the line-circle axiom holds, then an Archimedean H-plane must be either Euclidean or hyperbolic, but its coordinate field could be any Euclidean subfield of R, such as the field of constructible numbers or the field of real algebraic numbers.
See also my March 2010 MONTHLY survey paper mentioned by Will Jagy entitled "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries." Section 2 is all about Will Jagy's results about regular-polygoning circles in the hyperbolic plane (you can't always "square" them), and Section 3 is about the undecidability and consistency of elementary geometry.
If you email me at mjg0@pacbell.net I will send you my latest updating of that article.
(The terminology for all this is confusing. Yes, Janos Bolyai did introduce the term "absolute geometry" for the common part of real Euclidean and real hyperbolic geometries. I have argued - and it has generally been accepted by other writers - that "neutral geometry" is a better name, because one remains neutral about which parallel postulate to assume. I also argued that "absolute geometry" should be the name for Bachmann's geometry based on reflections, since it includes not only neutral geometry but also elliptic and other geometries - see p.588 of my book. Furthermore, even for neutral geometry, why restrict to real geometries? A model of Hilbert's incidence, betweenness and congruence axioms we now call a Hilbert plane or an H-plane for short. Pejas classified all H-planes. His classification is described on pp. 588ff of my book.)
Best Answer
The Unit Distance Problem asks:
A properly scaled square grid gives a lower bound of something like $g(n) \ge n^{1 + \frac{c}{\log \log{n}}}$, and a beautiful application of the crossing number lemma gives that $g(n) = O(n^{4/3})$.
A closely related problem where great progress was made very recently is the Distinct Distance problem, asking for the minimum number $f(n)$ of distinct distances among $n$ points in the plane. (Clearly $f(n)g(n) \ge {n \choose 2}$.)
Guth and Katz recently obtained a sharp exponent for $f(n)$. Terence Tao and János Pach wrote nice summaries of this work.