Q. “Neither I/me nor my dog responded.” Should it be “me” or “I”?
A. The correct form of pronoun in sentences like yours can usually be figured out by considering the pronoun separately from the noun that it’s paired with. In your sentence, that process would look like this:
I/me didn’t respond.
plus
My dog didn’t respond.
equals
Neither I/me nor my dog responded.
If your goal is to use Standard English, the correct form would be “I didn’t respond,” which answers your question: “Neither I nor my dog responded.” See also CMOS 5.44; for more on Standard English, see 5.4.