It doesn’t matter since he was, ethnically. That’s all that matters to the anti-Semites of Europe. Don’t think for a moment that was not a factor in the clash between Reds and National Socialists in the 30s.
Yes, but you mean “racially” I think.
Marx wrote a LOT, including lots of private letters. I’ve never seen any evidence of ethnic Jewish self-awareness on his part, except as would have been taught in his Lutheran upbringing.