This is a nice easy one.

Clear cache on the browsers. In IE 9 this can be hard, and sometimes requires you manually delete temporary folder files. This is caused by IE9 having a terrible development company behind it, namely Microsoft.

I have tested in IE9 and FF6, and Chrome. No problems in any of them, including the close panel feature. 

I have seen these strange things before, and always it is caused by cache doing strange things with dev work. Clear your cache and you should be gold!

Oh, sorry I had miss understood that you were clicking the login and register in your default skin links.

We found that because the results were unpredictable, we used some custom script in the jQuery custom script settings of the template to hide these.

If you add: 

jQuery("#dnn_dnnLOGIN_cmdLogin").css("display","none");
jQuery("#dnn_dnnUSER_cmdRegister").css("display","none");

to the custom script settings of the template, below the initial settings:

jQuery(document).ready(function(){
jQuery("#open").click(function(){
jQuery("div#adv_slidedown_panel").slideDown("slow");
});
// Collapse Panel
jQuery("#iwebs-close").click(function(){
jQuery("div#adv_slidedown_panel").slideUp("slow");
});
// Switch buttons from "Log In | Register" to "Close Panel" on click
jQuery("#toggle a").click(function () {
jQuery("#toggle a").toggle();
});


    jQuery("#dnn_dnnLOGIN_cmdLogin").css("display","none");
    jQuery("#dnn_dnnUSER_cmdRegister").css("display","none");

This hides the skin links and forces the user to click the slide down bar.

Logout should still work, as the slide down is only displayed for unauthenticated users. Try it.