Legally, he was eligible to run for President. I don’t know British law, so I can’t answer the second part. In the era in which he was born, it was very unlikely that he would have settled in America, since his father was a British nobleman. Moreover, his father started out as an influential MP. So, while he legally could have held both offices, a scenario in which he would have considered American politics is far-fetched.
Of course, had his mother divorced early she might have returned to America and then … ?
As a side-note, Churchill was invited to give a speech before a joint session of Congress. In the speech he quipped: “Had my father been American, and my mother British, I might have made it here on my own!”
Churchill didn’t regard himself as an American citizen. The president of the United States who conferred an honorary citizenship on him in 1963 didn’t consider him an American citizen.
No, the court ruled that Wong was an American citizen because he was born in the United States (neither of his parents was an American citizen). Churchill was born in Oxfordshire, England.