From Dave Sikula comes an identification of the clip in the previous posting…
In re, the Gene Kelly/Woody Allen clip, based on the poster for Half a Sixpence (which ran from April 25, 1965 to July 16, 1966) in Shubert Alley, I'm guessing we're looking at a bit of Gene Kelly: New York, New York, which according to this site and the IMDb, aired on February 14, 1966 with Julie Andrews as the other guest star. "This is not available on video, however, if you visit New York City or Los Angeles you can view it at the Museum of Television and Radio." Weird to see the two of them together. Good catch.
That sounds like it. Thanks, Dave.