The notion $f^{\circ 2}(x)$ for $f(f(x))$ is, while not common, pretty well-established in a number of references where there might be confusion with other uses.

Also, a caveat: you use the phrase 'a minimal surface' but it's not clear that there's a single minimal surface spanning the polygon and picking out the one of least area may not be trivial/formulaic.

What sort of formula are you expecting for your answer? Given that (as you note) the surface is likely to be very complicated, your best-case scenario is probably going to be getting the result in terms of ratios of elliptic functions of cross-ratios or somesuch, but all of the examples I've seen solved explicitly (see the 'four lines' surface at people.fas.harvard.edu/~sfinch/csolve/ge.pdf , for instance) seem to take essential advantage of some of the symmetries of the example.

The dragon curve was the first thing that sprang to my mind too. That paper is a great reference; thank you!

I don't think you need 'trails can be destroyed' - here's another more-precise formulation of the problem: each person has a finite number of (either directioned or directionless) objects, labeled $1$ through $n$ (and distinct from the other person's objects). Each person has some finite viewing distance $d$; they can see everything within $d$ and nothing outside it. What constraints on $d$ and $n$ allow the two to meet, and how does the meeting time depend on $d$ and $n$?

That's sort of my point, though - the fact that you can 'chain' repeated instances of the product-of-two-squares identities means that you can get arbitrarily many ways of expressing one sum-of-two-squares (for instance) as a product of two others, and it's not clear to me that a single parametrization can capture all of those distinct expressions.

For instance, taking the 'product' of the representations $5=1^2+2^2$ ($=\left|1+2i\right|^2$) and $13=2^2+3^2$ ($=\left|2+3i\right|^2$) yields the representation $65=4^2+7^2$, whereas using $13=3^2+2^2=\left|3+2i\right|^2$ yields $65=1^2+8^2$. Now you need to find a parametrization that yields both $(1^2+2^2)(2^2+3^2)=4^2+7^2$ and $(1^2+2^2)(3^2+2^2)=1^2+8^2$.

Is there any reason to expect a complete solution even for $a=b=c=1$? For instance, there are two distinct ways of writing $n=pq$ (with my $p$ and $q$ representing primes) as a sum of two squares if $p$ and $q$ can each be (i.e., if they're $\equiv 1\pmod 4$); this means that for a product of $k$ distinct primes there are $2^{k-1}$ ways of writing it as a sum of two squares, and many many ways of writing that sum as a product of two sums of two squares, and I can't imagine how you would parametrize all of them at once.

@JoelDavidHamkins That's a good point - there's no 'smoothing' happening here, essentially, so it's a 'cubical' and not a spherical surface that has minimum surface area. For $n=k^3$ this is bound to be best, and it's possible that you could show that it reduces to a 2d case for e.g. $k^3\lt n\lt k^3+k^2$.

In the limit, canonically these 'should' converge to their continuous analogues, but for specific values these sort of discrete geometric optimization problems are (at least to me) notorious for the lack of structure or symmetry in individual solutions, and I wouldn't necessarily expect a canonical answer.

It's certainly not random, but I don't know if I agree with the last assessment; I find it quite interesting. There's actually a rather deep recursive structure in the order of the coefficients that's related to the continued-fraction expansion of $\frac pq$ (the best references I know of for this behavior come from discussions of Bresenham's line-drawing algorithm, but that may just be because it's how I was originally introduced to the topic).

As Wlodzimierz notes in their answer, the 'converging sequences of shapes converge to non-overlapping limits' seems misguided - what non-overlapping convergent limit would you expect to get out of $\{[0,1]^2\cup [0,1]\times[1+\frac1n, 2+\frac1n]\}$ as $n\to\infty$?

Something feels wrong here - in (1) I presume you mean $f(k)=a_k$, and in this case the polynomial doesn't always have integer coefficients - for instance, as you note, ${n\choose 2} = \frac{n(n-1)}2$ is always an integer, but its coefficients certainly aren't integers.

The next natural question to ask seems to be density of attracting basins: what are the asymptotics of (for instance) $\frac1N\left|\{x: x\leq N\wedge f(x)=5\}\right|$?

@GHfromMO Mr. Hardy? Is that you?