(Split in to two parts as it will not let me post more than 1,000 char)

Great question. I will do my best to address each part of your question in some detail.

First, I would like to address the issue of her having an "equivalent reputation in the U.S." Among African Americans and some whites, Josephine Baker absolutely had a stellar reputation in the United States. Pages of African American newspapers from the Atlanta Star to the Chicago Defender to the New York Amsterdam News tell of her triumphs on the stage. The New York Times also covered her, and by 1927 she reported receiving 40,000 love letters and over 1,000 marriage proposals. {"The Legend Named Baker Comes Home," New York Times, February 2, 1936, X2} Even if only half-true, that is still a tremendous amount of people who adored Josephine Baker.

As for films, while she liked films, she never had any plan to make herself a film star, preferring the stage. {ibid}

On Josephine Baker's view of racial discrimination in the United States, there are some published examples I can point you to. She claimed her hometown of St. Louis was "one of the worst cities in America for racial discrimination." {Murray, James P., "Josephine Baker Dies in France," New York Amsterdam News, April 23, 1975, p. A1}

In terms of how she felt about racial equality in France, a very interesting quote emerges from an interview in which she was asked by the reporter if she would ever return to the United States permanently:

"Yes, I would come, if I can enjoy the same feelings and and live the same type of life that I do in France." {Booker, James, "'I'd Come Home--If'" New York Amsterdam News, Feb. 20, 1960, city ed., p. 1.}

Her 1963 March on Washington speech also tells of the stark differences she experienced in Europe versus the United States (as a bonus, absolutely beautiful oration):

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But I must tell you, when I was young in Paris, strange things happened to me.  And these things had never happened to me before.  When I left St. Louis a long time ago, the conductor directed me to the last car.  And you all know what that means.

But when I ran away, yes, when I ran away to another country, I didn’t have to do that.  I could go into any restaurant I wanted to, and I could drink water anyplace I wanted to, and I didn’t have to go to a colored toilet either, and I have to tell you it was nice, and I got used to it, and I liked it, and I wasn’t afraid anymore that someone would shout at me and say, “N-----, go to the end of the line.”  But you know, I rarely ever used that word.  You also know that it has been shouted at me many times.

So over there, far away, I was happy, and because I was happy I had some success, and you know that too.

Then after a long time, I came to America to be in a great show for Mr. Ziegfeld, and you know Josephine was happy.  You know that.  Because I wanted to tell everyone in my country about myself.  I wanted to let everyone know that I made good, and you know too that that is only natural.

But on that great big beautiful ship, I had a bad experience.  A very important star was to sit with me for dinner, and at the last moment I discovered she didn’t want to eat with a colored woman.  I can tell you it was some blow.

And I won’t bother to mention her name, because it is not important, and anyway, now she is dead.

And when I got to New York way back then, I had other blows—when they would not let me check into the good hotels because I was colored, or eat in certain restaurants.  And then I went to Atlanta, and it was a horror to me.  And I said to myself, My God, I am Josephine, and if they do this to me, what do they do to the other people in America?

You know, friends, that I do not lie to you when I tell you I have walked into the palaces of kings and queens and into the houses of presidents.  And much more. But I cold not walk into a hotel in America and get a cup of coffee, and that made me mad.  And when I get mad, you know that I open my big mouth.  And then look out, ‘cause when Josephine opens her mouth, they hear it all over the world.

So I did open my mouth, and you know I did scream, and when I demanded what I was supposed to have and what I was entitled to, they still would not give it to me.

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A 1951 incident involving New York's Stork Club and Baker actually led to changes in civil rights ordinances in New York City.

Baker alleged she as being discriminated against. Most sources claim Baker was refused admittance; years later in her obituary, it is documented as after "an evening of delayed and indifferent service." {Murray, James P., "Josephine Baker Dies in France," New York Amsterdam News, April 23, 1975, p. A1}

The ruckus that resulted from the Stork Club refusing to serve Baker and the police refusing to assist actually resulted in the passage of some public accommodations non-discrimination ordinances by the city council there. The ordinance allowed the police to revoke the licenses of public establishments that discriminated against patrons based upon their race. {Jackson, R. "PULSE OF THE PUBLIC: Granger Thinking Always Against Negro," New York Amsterdam News, October 9, 1954, city ed., p. 16} This is actually a concrete civil rights action Baker's Stork incident was responsible for.

After the Stork incident she was lambasted in some circles. She ended up threatening to sue columnist Walter Winchell after he "turned against" her activism. {Murray, James P., "Josephine Baker Dies in France," New York Amsterdam News, April 23, 1975, p. A1} This was after the press claimed she was on the side of Mussolini against the Ethiopians in the 1930s. It appears the story may have been based on a prior quote of Baker's, but she later denied she'd said it, some press said she did. {"Winchell Wrote," New York Amsterdam News, Sept. 17, 1955, city ed., p. 16.}