Why some rules in Elementary Cellular Automata halt and others don't?
Your problem is formalized in research as follows: given a CA, is it "Turing complete"? This has been proved for a limited set of CAs including rule 110 in the early 2000s by Cook in a notable proof. it was probably first proved for a CA for the game of life by Conway in the 1970s. There are empirical ways of analyzing CAs that can rule out Turing completeness, and "signs/ circumstantial evidence" that a CA might be Turing complete, but no CA can be proved Turing complete or not from a finite number of example cases.

TMs can execute an arbitrary number of steps and then halt, so that is a characteristic required of the CA simulation. But there is no standard way of defining whether a CA "halts" and thats up to the universality mapping. There is currently no general theory of CA←→TM mappings. There are different mappings in the literature and Cooks mapping is also called "weakly Turing complete" because it requires infinitely recurring patterns specified on the CA input.

There is no general way to find whether particular CAs and rules are universal. This is essentially equivalent to the halting problem in general.

Unfortunately Wikipedia does not highlight Turing completeness in CAs. Here are two useful alternative references.


Universal Cellular Automaton / Wolfram Mathworld by Weisstein
Universality in Elementary Cellular Automata / Cook