I'm trying as autodidact to read chapter 4 of Foundation of Cryptography by Oded Goldreich (just to let you "tune" your answers, I have engineering background).
If I'm correctly understanding, giving a perfect simulator $S_1$ the possibility to halt is not a problem because we can define a simulator $S_2$ which repeats $S_1$ let's say $n$ times, outputting the result of the first not-halting $S_1$ iteration, or a "dummy" result if ALL $S_1$ iterations halt. This way the probability of $S_2$ outputting the dummy result can be lowered as liked with the growth of $n$.
$S_1$ halting probability is bounded above by $1/2$, but why? It seems to me that every $S_1$ halting probability $<1$ will be lowered towards $0$ by a sufficient large $n$. More, the simulator one seems a very different argument from completeness/soundness probabilities, where the strict $1/2$ threshold is justified by the majority rule applied to that (different) repetitions strategy.
And, btw, is there any reason to choose $S_1$ repetitions value $n$ to be the same as the other repetitions number needed to pass from weak completeness/soundness to stronger ones? Or are the numbers of the two kinds of iterations mutually independent? I guess this doubt comes from me being confused about if $S_2$ is the simulator for the weak IP, or for the stronger IP...