Interesting post. Some comments:

• There are also some nice demonstrations including Reverse Collatz Paths.

• Assuming that you've tested all values below n for loops or convergence to 1, you can stop applying the Collatz map as soon as you get below your starting value, e.g.,

CollatzStop[m_]:= NestWhileList[n |-> If[EvenQ[n], n/2, (3 n + 1)/2], m, p |-> p >= m]

• The first n+1 terms in the Collatz iteration starting with (2^n-1) are increasing so one can (easily) generate Collatz sequences of arbitrary length.

• There is a simple relationship between the length of the Collatz sequence and the (log of) primorial starting values.

• The Golden Mean Shift is the Set of 3x+1 Itineraries was a poster presentation at the 2004 NKS conference! In particular, Section 3 A Mirror Function On the Unit Interval presents a relatively simple recursive figure.

• You might find the last section of the slides of Encoding Challenges for Hard Problems (Heule 2019) interesting.

• Collatz's "3x+1" problem and iterative maps on interval relates Collatz to a (rational) map in [0,1], which looks interesting. Perhaps this could be turned into a good Demonstration?

• Richard Crandall was interested in this problem for a long time, e.g. in On the "3x + 1" Problem

That is a very nice and interesting outline of the Collatz conjecture - thanks for sharing! Just as a shy remark: I find it inspiring if the above "scatter plot of seed vs. sequence length" is plotted using logarithmic scaling (ScalingFunctions -> {"Log", Automatic}). This way you get a surprisingly regular pattern:

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