Just to clarify that my quotation
"Nevertheless, any Polish space is the continuous image of the Baire space."
is not contradicted by your observation that
"not every Polish space can be mapped continuously onto the Baire space".
You are right. I initially thought it is not possible to map irrationals continuously onto reals in such a way so the image is the whole real line, but it turns out it is possible: https://mathoverflow.net/questions/112127/the-reals-as-continuous-image-of-the-irrationals/112130
I do not get it. It seems to me that it is a forest since there are several possible substitutions at every string.
I am not sure I understood your substitution system correctly then. Could we say, that on each step you are adding a number to the end of a string? Or, rather, you are inserting a number into any place of the string? If the first case is true, then the causal graph for any evolution history will be a single chain, since at each step you are using the output of the previous step (and thus each update is connected to the previous update). If the second interpretation is true, then the causal graph can be pretty complex, but does not have to have a tree structure (can still be a chain like in the first interpretation though).