I want to make a block that shows a poll that shows the results and with a link to vote to that particular poll. It seems that the default most recent poll shows the results if the user doesn't have the right to vote and the voting options if it has the rights...
How can I control this?
From memory you can configue poll blocks using the Views module. At any rate, using Views you will be able to create a new block that displays votes. You can then add a footer to the block with a link to the poll node so users can vote.
There's plenty of documentation about using Views: