The bizarre show that Roger and Jane put on is a "blackface" number. Blackface, in which a white (or, much more rarely, a black) performer would paint his or her face black and sing and dance in a way that was stereotypically (though rarely correctly) associated with African Americans, was a hugely popular performance style during the nineteenth century and into the first part of the twentieth. But during the time this episode is set (1963), it was starting to fall out of fashion somewhat, especially as awareness of racial inequalities and stereotypes rose among white Americans.
The Drapers live in Ossining New York on the Hudson river. Infamously known for the Ossining NY State Correctional Facility, known in local slang as Sing Sing. The expression is used often in old movies such as Breakfast at Tiffanys and Angels With Dirty Faces. Another NY coloquialism referring to Sing Sing is "Being sent up the river" which meant going to jail.