Beth definitely sang "Not a Day Goes By" for more than two weeks. I'm not sure exactly when it was given to Frank, but Beth sang it for at least two-and-a-half weeks, maybe more than three. It's possible that one point it was given to Frank for a few days, then it was given back to Beth, and then back to Frank again.
At the moment, I don't have the time to respond in a lot of detail about my feelings on all these questions, except to say that I think Beth should sing it. I think it was a mistake to take it away from her. I have heard that it was taken away from her because some people felt that Sally Klein didn't sing it well enough. If that was the reason, I think it was crazy. And so does most everyone I know who saw Sally Klein do the song. Maybe her singing wasn't thrilling, but it was good enough, and her acting of the song was so beautiful. Her acting was beautiful throughout. For me, no later Beth I've seen has come close. She was just so likable up there, so endearing, so simple, real, funny, delightful and touching. Has anyone been nearly as funny in "Bobby and Jackie and Jack"? No (or at least no one I've seen).