You are using 'me' as the subject. It can only be used as the predicate or indirect object. To see this, take away 'myself'. You would have "Me also made the same mistake" - doesn't sound right, does it? You could say "As for myself, I also made the same mistake", which is another way of saying "Considering me in relation to the subject, I also made the same mistake."

No. It doesn't sound quite right. You could say: "I myself, also made the same mistake" is a better statement.

If you are the subject - start the sentence with I not me. "I also made the same mistake." or "I myself, also made the same mistake."