Chat window hiding behind Twenty Eleven header

It is indeed a z-index problem. The #branding element has a z-index of 2, in other themes (like Twenty Ten, the default in 3.0 and 3.1), it was not set.

A simple solution is to set the z-index of #chat-block-site to something higher.

#chat-block-site
{
    z-index: 10;
}