We use "half" as "a mixed race. In this case it means American-Japanese or Japanese-American. (I wonder if these words are correct in English.) "I'm a mixed race of America and Japanese but I can speak only Japanese." Sorry that my English isn't good.

"で" in this sentence indicates the condition of "being half Japanese and half American" and could be translated into "and" For example, 私は60歳で未婚です。（I'm 60 years old and I'm unmarried.） 彼女はアメリカ出身で英語を教えています。（She's from America and she is teaching English.） 彼は日本人で英語が堪能です。(He is Japanese and he is fluent in English.) 「日本とアメリカのハーフで日本語しか話せない」= The person (the pronoun is unclear, because it should be determined by the context.) is half Japanese and half American and can speak Japanese only. Hope this helps.

You are right and correct. Other way to say Nihonjin to Amerikajin no half dakeredo Nihon-go shika hanase masenn. is otherway