Lowerbound for the minimum distance between points in a "nice" triangulation
The short answer is: no, there is no such lower bound.

Let $p$ be a positive integer.
Let $\epsilon = \frac{1}{p}$.

Put points at coordinates $(i \epsilon, j \epsilon)$ for $i,j \in \{ 0 \dots p \}$, and at coordinates $(i \epsilon + \frac{\epsilon}{2}, j \epsilon + \frac{\epsilon}{2})$ for $i,j \in \{ 1 \dots p \}$.
You end up with $n = (p+1)^2 + p^2$ points, and a triangulation in which all triangles are both right and isosceles. So all angles are either $\frac{\pi}{2}$ or $\frac{\pi}{4}$, and the minimal distance between two points is $\frac{\epsilon}{\sqrt{2}}$.