Did Rand rely on Paine's research and comment, or did they independently come to many of the same conclusions?
To my knowledge, Ayn Rand
came somewhat independently to her own views. She may have imbibed Paine second-hand through such Old-Right acquaintances and influences as H.L. Mencken, Henry Hazlitt, and Isabel Paterson. As to the influence of the last, Paterson scholar Stephen Cox will be discussing the relationship between these two thinkers at our Summer Seminar conference this July.
It does not appear that Rand formed her views from close prior study of Paine or other American authors. But she did read some works, and she does refer approvingly to the Founding Fathers in her writings. That she understood their views in essentials may be due to their lasting influence in American culture and discourse, and their presence in secondary literature and even journalistic accounts that she may have read or discussed.
Her only reference to Paine comes from the 1940s, in a letter she wrote to Leonard Read, founder of the Foundation for Economic Education. It is not approving. Ayn Rand wrote that Paine was "not one of us" (Letters of Ayn Rand, p. 171), i.e., not a defender of the free market. In this she presumably refers to Paine's embrace of the French Revolution. (In that same letter, she championed Paterson's The God of the Machine.)
She approvingly edited and published an excerpt from Leonard Peikoff's The Ominous Parallels that cites Paine. It is in "America's Philosophic Origin" (in The Ayn Rand Letter, Vol. 3, No. 5) in the following paragraph: "The result of the Enlightenment ideas and attitudes, in every branch of philosophy, was a surging sense of liberation. 'We have it in our power to begin the world over again,' says Thomas Paine." (And the quotation from Paine continues.)
In the Objectivist view, Paine's own inconsistencies reflected the inconsistencies at the heart of Enlightenment philosophy. These have been exploited to devastating effect by critics of the Enlightenment, and indeed that process was bloodily evident even in the French Revolution. If we are to recapture the spirit of the best in the Enlightenment, we need to revise that philosophy as much as we need to pick and choose essential principles from it.