This sort of addresses the question in your last paragraph, but it is actually a bit tangential. Hopefully it is still of interest.
It is well known that every planar graph has an embedding such that each edge is a straight line segment. It is therefore natural to ask if every planar graph has an embedding in $\mathbb{R}^2$ such that the length of each edge is integral. This was first conjectured by Kemnitz and Harborth. It was proved for cubic planar graphs by Geelen, Kuo, and McKinnon here.
Edit: After reading the paper a bit more carefully, I see that it is more related to the question at hand than I initially thought. Apparently, Erdos posed the following question:
How many points can we find in the plane with pairwise rational distances such that no three are on a line and no four are on a circle?
A collection of seven such points has been found by Kriesel and Kurz, but it remains open whether a collection of 8 such points exists.
Also, Theorem 2.1 of the paper, proven by Berry seems to be of interest.
Theorem. Let $A,B,C$ be non-colinear points of $\mathbb{R}^2$ such that $d(A,B), d(A,C)^2$, and $d(B,C)^2$ are rational. Then the set of points of $\mathbb{R}^2$ that are rational distance from each of $A,B$, and $C$ forms a dense subset of $\mathbb{R}^2$.
Theorem 2.2 of the same paper seems to be quite pertinent as well.
