Why every rational langage is the image of a local automaton
(1) Indeed. Every regular [=rational] language is the morphic image of a local [=2-testable] language.

This is seen as follows. A finite state automaton can be "encoded" by a 2-testable language by using the transitions as letters. So we consider symbols $(p,a,q)$. The corresponding language is 2-testable: basically test whether the transitions are consecutive which is clear from letter pairs: $(p,a,\underline q)(\underline q,b,r)$. We find the original language by applying a coding: $(p,a,q)$ maps to $a$.

(2) No. Local languages are not closed under codings. $(ab)^*$ is a local language. Initial letter $a$, final letter $b$, two letter substrings $ab, ba$. The image $(aa)^*$ is not local. There is no way to distinguish odd length strings from even length strings by just looking at substrings.