How to change the order of the comment_form fields with css?

You could use flexbox instead of table if wanting to do this with css. Flexbox lets you order elements easily:

#commentform {
    display: flex;
    flex-flow: column;
}
.comment-form-comment {
    order: 1;
}
.comment-form-url {
    order: 2;
}
.comment-form-email{
    order: 3;
}
.comment-form-author{
    order: 4;
}
.form-submit{
    order: 5;
}

Here’s an example:
http://codepen.io/candid/pen/NxKJyW/