Thanks, Joe. I think the rest of it is right, at least in the sense that it is a faithful transcription of one of Lehrer's recordings of the song. On another recording (the earliest, I think), he had Ingrid Bergman playing part of hypotenuse. (Who would he name if he were recording it today?)

The two places where it says "Russian double-talk", you can find webpages where people have transcribed what he said on the recordings --- but unfortunately these people don't agree with each other as to the transliteration or the translation, and I don't know any Russian so I can't give an informed opinion on the issue. So I give you this with no assurance that it's right:

The first is "Zhil byl korol' kogda-to, Pri njom blokha zhila," which means "Once upon a time a king had a pet flea," which Mephistopheles sings in Goethe's Faust, in the Russian translation composed by Musorgsky.

The second line is "Ja edu kuda sam tsar idet peshkom", which translates as "I am driving where the king himself goes on foot", which hints at the toilet, but actually means that someone is trying to outdo the king in a little unimportant matter, and is being rather foolish at that.