I can't give a complete answer, but I can make a few observations in the general direction of an answer. Note that most of this is based on memory from spending several hours looking at the behavior of these plots and observing how they relate to roots for individual polynomials. This is all "empirical math" (assuming that's even a thing, heh) but the patterns are pretty clear and while I doubt I have the mathematical background to make most of it rigorous I doubt it would be too difficult.
It's not limited to coefficients from $\{-1,1\}$; almost any collection of polynomials with coefficients chosen from a very limited set will produce similar patterns.
The character of the pattern is directly tied to the location on the complex plane, specifically the effect of multiplication. So I suspect that it catalogs the IFSs only insofar as the complex plane catalogs (some) affine transformations. Note that the Julia set formula, in contrast, lets the transformation vary during iteration with only an additive constant to alter the shape, which is why it has the more elaborate chaotic behavior vs. the IFS-like structure here.
Each fragment of a pattern roughly resembles a Cantor set, with fragments overlapping heavily except at the outer fringes. Each fragment is twisted according to the nature of complex multiplication, and the combination creates the similarity you noticed.
The latter two points are fairly intuitive given the nature of complex numbers and mostly apparent from inspection of the plots, but don't really do much to explain why, which I imagine is why Baez didn't remark on it.
Also unexplained are the gaps, such as those around the roots of unity. I don't recall the exact placements, but I recall it also being obvious why the locations of the gaps were relevant, but not why root density drops off so sharply around them. I could probably make some guesses, but only with copious handwaving involved.
On the other hand, except for the gaps, I believe the thick ring around the unit circle simply consists of more variations on the same patterns, denser and heavily overlapped, until detail is no longer visible.
Regarding variations on plots like this, the same approximate pattern appears regardless of polynomial degree, and for any set of coefficients of the form $\{-N, -(N - 1) ... -1, 1 ... N - 1, N\}$. The density of roots around the unit circle, the prominence of the gaps around certain points, and the character of the fringe inside the gaps all vary with degree; this is hard to see in Derbyshire's plot rather than one limited to polynomials of a single degree.
The shape of the overall plot changes if the coefficients are not chosen as above, being either of different magnitudes or chosen from more than one set. Note the plots on this page, particularly those near the bottom. The coloring indicates sensitivity of the points to changes in coefficients, which indicates which parts of the plot may distort asymmetrically.
My observations, in general, were:
Doing both in moderation creates... odd effects:
I always think a discussion of the Mandelbrot set should go along with a discussion of the logistic family, as a simple (the simplest?) interesting model of population dynamics. Of course the logistic family is only the real part of the Mandelbrot set, and here is a first simple pre-calculus exercise: show how to change coordinates to get one parametrization from the other.
(Or: Even show that you can change coordinates for every complex quadratic polynomial to get one of the form $z^2+c$.)
The fact that the Mandelbrot set is bounded was already mentioned in a previous answer. It's easy and straightforward, and well worth covering.
Some other easy exercises would be to determine the fixed points of the polynomials, and the region where there is an attracting fixed point (derivative of modulus less than one), i.e. the "main cardioid" of the Mandelbrot set. Do the same thing for period 2, and maybe ask the question what happens with higher periods - see also comments about density of hyperbolicity below.
It is also possible to discuss the structure of the Julia set (phase space), and in particular the difference between disconnected Julia sets outside M and connected Julia sets inside. While a formal proof would be difficult at this level, giving the geometric idea is not too hard, and for very negative real c (i.e. large $\lambda$ in the logistic parametrization), the result that the invariant set is a Cantor set is easy to do with elementary means (this is done e.g. in Devaney's book "A first course on chaotic dynamical systems").
You say that connectivity of the Mandelbrot set might not be interesting to them, but it's always possible to tell the amusing story about how Mandelbrot's first computer pictures suggested that M is disconnected, but the editor of the paper carefully removed all the little islands from his picture, thinking they were dirt! You could accompany it by running two different algorithms (one which just does a pixel-by-pixel calculation and colors points white or black, making M look disconnected, and e.g. the more standard colored pictures that show the connectedness of level lines quite clearly), and thus making a point about the perils of computer experiments and the importance of mathematical proof (if you care about this).
Finally, I would say that it is a good idea to talk about density of hyperbolicity. The question whether every interior component of the Mandelbrot set corresponds to maps with an attracting cycle is one of the most important open questions in complex dynamics, and yet is quite easy to understand with just a little bit of experimentation. It is always good to show students that even seemingly innocent questions can be the subject of very difficult mathematical research. Of course this is also a chance to mention that density of hyperbolicity in the real case (i.e. density of period windows in the bifurcation diagram) was only established in the 90s, and was a major mathematical breakthrough.
I realize there is a lot here, and it may go beyond what you were looking for, so pick and choose!
Best Answer
Maybe dense parts of the parameter plane
In generally one can zoom in infinitely many places which takes time ( limited !) and precision, so there are infinitely many such places. See also perturbation method for some improvement.
Similar interesting problem is on the dynamic plane : there are some Julia sets ( Non-computable Julia sets ) which were note yet been seen graphically ( even without any zoom) : Cremer Julia sets