Translation:You wrote this character incorrectly.
I'm not sure about your first sentence, but I don't think so. I think that your second and third sentences are grammatically correct, but I don't think I've ever heard a Chinese person use 错 that way. In my experience, 错 is usually just put directly after the verb as a verb complement, or appears on it's own (meaning someone was incorrect or made a mistake/did something wrong).
It's not wrong for Duo to teach that 字 = "Chinese character" because that is probably the most common meaning; however it also means simply "character" (of any language) and also "word" (of any non-character based language) so without context both of these should also be accepted.