The Random Post Box plugin places a box anywhere on the blog, where it loads random posts one-after-the-other.
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.
The box is a div with id "random-post-box-frame", and the markup inside is quite similar to the default theme.
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.