I gave your questions to my cousin who has a fantastic knowledge of old songs and films but some of the clues are too vague and could apply to lots of songs according to him.
He did come up with an answer to No.4 but he said it's such an obscure song from 1926 that he doesn't think many other people would know it. However, I'll offer it to you.
The song is called I'd rather Charleston.
The first line is 'Take a lesson from me and the reply is I'd rather Charleston'
I should have thought of "I'd Rather Charleston" - I don't think it's all that obscure, at least if you're into old Hollywood musicals. Thanks for reminding me of it, FoxLee2!