I bought my first game, and didn't get my bonus code for the free game. I've read around and I realize now that you had to have your betacode email as your email for the account. (Which I don't - Hence the lack of a free code).
However, if I change the email address to the one from the beta code, and buy another game will I be sent the free game code? (This would be my second purchase).

Quoting from the promotion rules: "You have to register an account at GOG.com with the same email address as the one you've signed up with"
Seeing that you didn't I don't think the promotion will work after you change your email address. Wait for an official response though or send GOG an email (it may be faster).