The Convention would be for the purpose of proposing amendments. I suppose they could propose whatever amendments they want, but they cannot wholesale write a new Constitution. That said, I suppose they could try to do what the Senate does with House bills and pass a single amendment that says "Strike out the entire Constitution and replace it with this..." I doubt that an amendment proposal like that would win a vote at the convention, and even if it does, I'd expect 13 states to reject it.
The states wouldn't cherry-pick amendments, they would vote on each separately, yea or nay. The current Bill Of Rights were voted on separately, with different combinations of states voting for different amendments.
-PJ
Could the Convention set a time limit for the State Legislatures to debate their proposed changes? When the Congress sent the ERA Amendment to the legislatures for debate it stated that if the Amendment was not ratified by a certain date the proposed Amendment would die.