Does a language stay regular if you re-encode the input?
The answer is yes. This is generally called homomorphism, and regular languages are closed under homomorphism. Moreover, regular languages are closed under inverse homomorphism also. More precisely,
if $\tau$ is a homomorphism from alphabet $\Sigma$ to alphabet $\Delta$ and $L$ is a regular language over $\Delta$, then $\tau^{-1}(L)$ is also a regular language.

You can take a look here for more details, or the best is to take a look at this book, which contains proofs about basic properties of regular languages.