I recall this one quite well.
(2) It takes place in their apartment
(3) It's pretty good. Another great example of classic Ernie and Bert.
(4) The ending music was the same used in quite a bit of skits, including non Ernie and Berts, such as the skit where Cookie Monster rings the bell at the carnival and wins a cookie.
(1) Bert is holding a cookie in his hand and is happily about to enjoy eating it. Ernie comes along and asks Bert if he will share it with him. Bert refuses. Ernie persists in trying to persuade Bert and even physically holds his arm to prevent Bert from taking a bite! Ernie says, "If it was my cookie, I'd share it with you." Bert says, "No, you wouldn't." Ernie then takes the cookie from Bert. Bert gets mad and demands it back but Ernie says, "Wait Bert let me show you. Pretend this is my cookie. Now you ask me if I'll share it with you."
Bert thinks this is silly and wants the cookie back, but Ernie persists so Bert finally complies and says, "Ernie, will you share that cookie with me?"
Ernie then says, "Why of course,Bert." He breaks it in half, hands Bert one half, and then eats the other. He walks away, and Bert stands there dumbstruck and says, "I don't get it". The end music is playing at this point. But the skit isn't quite done yet! As soon as the music ends, Ernie then reappears on the screen and says, "Hey Bert will you share that half cookie with me?" Bert then gives out a HUGE exasperated scream! It makes Ernie put his hand over his mouth (Kind of like he realizes he went a little too far with his ol' buddy Bert this time!)