WordPress.org

Is there an option to display several posts?

Not at the moment, but it might be included in future versions. It's not recommended to use the template tag several times on the same page.

How can I style the box?

The box is a div with id "random-post-box-frame", and the markup inside is quite similar to the default theme.

The box changes size when loading a new post

You need to set the size to something static with CSS, or use some means to control the size. Use the id "random-post-box-frame" as described in the previous answer.