Is it the case that:
Overall question: Does precision in some way refer to or depend on the density of numbers you can represent (accurately)?
Representable floating point values are densest in the real number line near zero?
In a full implementation of IEEE 754 floating point yes.
However in systems that do not support subnormals, there is a gap around zero which is substantially larger than the difference between the smallest nonzero value and the second smallest nonzero value.
Representable floating point values grow sparser (exponentially?) as the number line moves away from zero?
Yes, each time the value passes a power of 2, the gap between adjacent values doubles.
If the above two are true, does that mean there is less precision farther from zero?
That depends on how exactly you define "precision", one can talk about precision in either a relative sense ("significant figures") or an absolute sense ("decimal places").
Which is more appropriate depends on what exactly the numbers are used for. Loss of precision when moving away from zero tends to become a real concern if floating point numbers are used for things like coordinates or timestamps.
