Did the Soviets use the the Black Power, Gay rights, Women's rights, etc. movements in the 60's as propaganda against the U.S.?
The Soviet Union often brought up the mistreatment of African Americans as a way to point out flaws in a capitalist Democracy, and point out that Soviet domestic abuses were not dissimilar to the atrocities being committed in the United States.
Similarly, during the Korean War a significant amount of propaganda leaflets were written with the explicit intention of convincing African American soldiers to defect.
Yes very much. Soviets very much used the treatment of black people in the USA in propaganda, to show that the "freedom"-image of the United states was made up, and not based on reality. This was common for a long time, for example everybody's favorite revolutionary Angela Davis visited the Eastern Block and the Soviet Union several times.
But of course this wasn't accepted without an answer. While the Soviets accused the USA of racism, the USA accused the Soviets of antisemitism. I had some European newspapers that reported on these accusations and published them basically side to side to each other.
Another "funny" thing is, that these accusations actually weren't made up. I don't have to tell you much about racism in the USA, I think it is known enough here. Antisemitism in the Soviet Union was very real. Especially under the late Stalin in the 50ies antisemitism was awful in the Soviet Union. But also later on it wasn't all fine and dandy for Jews. There's a reason why there are next to no Jews living in the former areas of the Soviet Union and these aren't just economic reasons.
For the other areas asked, I think the womens rights questions were very much propagandized in the GDR. Objectively women were more equal in the GDR than in West Germany. One of the lonely achievements of the GDR.
edit: And gay rights weren't talked about, as the Eastern bloc was themselves discriminating gays. Arguably even worse than in the West. One of the last countries out of the Soviet bloc surviving, Cuba, only a few years ago said that this policy was wrong, and even these mostly because Castros daugther was pushing for acceptance of homosexuality in Cuba.