To my knowledge the expression「私なんか」is approximately translated as "Oh, not me!" and used to sound humble.
The questions are:
- Is it gender neutral?
- Are there other situations where it can be used?
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1.Yes, it's a gender-neutral expression. When used by a man, you can change the first person to mean the same thing, as in "俺なんか" or "僕なんか".
2.Depending on the sentence that follows it, it can sound pessimistic. When you use "私なんかダメな人間だ" you are saying that you can't see yourself as a good person.