The March Hare did not want Alice to join the party because he thought it to be uncivil to sit at an occupied table without an invitation. Although there was a chance of being beheaded, Alice possibly played the game of croquet in order to speak with someone else on her quest for knowledge about Wonderland. If you look at how the Queen of Hearts is, perhaps Alice played the game of croquet in hopes that if she participated, she would not get beheaded. I believe there could be an array of answers to that question. The reason why the cards were painting the roses is because they were not red and they needed to be in order for them [the cards] to keep their heads. The croquet game is played in a disorganized manner because the inhabitants of Wonderland are mad. In fact, I think that Wonderland itself is just a mad world. The cat on the cover of the book is portrayed in a disappearing fashion because it is the Cheshire Cat and he has the ability to disappear. I think the Cheshire Cat is on the cover of the book because he may play a bigger role toward the end of the story. Though I do not know for sure.