It's easy to form sets of five mutually-tangent spheres (say, three equal spheres with centers on an equilateral triangle, and two more spheres with their centers on the line perpendicular to the triangle through its centroid). Based on this, I think it should be possible to construct a set of spheres analogous to the Moser spindle [http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem] that requires six colors: spheres a, b, and c, where a and b have four mutual neighbors that are all adjacent to each other, a and c have another four mutual neighbors that are all adjacent to each other, neither a and b nor a and c are adjacent, but b and c are adjacent.
I have no idea how tight this lower bound might be, but it's at least better than four.
This does not really answer your questions, but I recently got a few results on the chromatic number of the hyperbolic planes. They are formulated by fixing the curvature and letting the distance vary, and I use the notation $\chi(\mathbb{H}^2,\{d\})$ for the chromatic number of the distance-$d$ graph on the hyperbolic plane with curvature $-1$.
for small $d$, $\chi(\mathbb{H}^2,\{d\})\leq 12$ (this can probably be improved, but maybe not easily to $7$),
for large $d$, $\chi(\mathbb{H}^2,\{d\})\leq \frac{4}{\ln 3} d + O(1)$.
The proofs can be found here: https://arxiv.org/abs/1305.2765, published in Geombinatorics Vol XXIV (3) 2015, pp. 117-134 (but the proof of the linear upper bound has some small issues, corrected with an improved bound in the subsequent work of Parlier and Petit https://arxiv.org/abs/1701.08648). All this is not difficult, and the paper raises more questions than it answers.
My impression is that the monotony of the chromatic number with $d$ seems reasonable, but is in fact a subtle issue; and I would rather bet on a negative answer to question (2) but not too high. All in all, these questions are probably incredibly difficult, because we have only very cumbersome tools to relate the geometry with the distance graph.
For the story: about one year after I read, liked and bookmarked your question, I had forgotten about it but read "Ramsey Theory, Today, and Tomorrow", and realized I could answer some questions asked in it by Johnson and Szlam. In the course of writing a paper from these answers, I investigated the case of the hyperbolic plane. After writing a first version I happened to look at my MO favorites, and saw your question again -- which is therefore cited in the paper (will there soon be a @mathoverflow in standard bibTeX definitions?)
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As of this morning there is a paper on the ArXiv claiming to show that there exists a 5-chromatic unit distance graph with $1567$ vertices. The paper is written by non-mathematician Aubrey De Grey (of anti-aging fame), but it appears to be a serious paper. Time will tell if it holds up to scrutiny.
EDIT: in fact, it must be the one with 1585 vertices, according to checkers, see https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/