OK I believe that with the help of the answers provided above and some additional research on my own I have answered my question.
There are actually 2 parts to this problem:
1.) Transparency issues with the .png close button in IE - This part was answered by Pointy above. According to Pointy "The browser is incapable of managing the compositing of the transparent PNG concurrently with any opacity effect on the image. In other words, for any given pixel, you either get alpha channel effects from the PNG itself, or from an opacity filter, but not both." (Quote taken from his answer at IE7 and "jaggies" around layered PNGs (with jQuery))
What I want to happen is for the .png close button to be semi-transparent in it's normal state, and full opacity in its hover state, but apparently this is not possible to achieve in IE using only one single image and CSS transparency. So to get around this I'm just going to have to make a dual-state .png image to get a nice hover effect, rather than relying on CSS transparency to handle that component.
2.) Transparency issue with the "reopen" button - This button should be semi-transparent but did not work in IE. After doing some extensive research on my own I came across this post: jquery IE Fadein and Fadeout Opacity. woodstylee's answer in that post solved my problem.
Due to some weird IE bug that I don't completely understand, I need to declare the element's opacity using jQuery/javascript before any transition/fade effects are applied.
Obviously there is some kind of conflict with jQuery fades/transitions and CSS opacity in IE. If anyone else has some additional info on this subject please post!
When I updated my code with the following, I was able to successfully get my "reopen" button to appear semi-transparent in IE.
$('#hello_mini').css('filter', 'alpha(opacity=75)');
Note that the above line of code needs to be placed before this line:
$('#hello_mini').show('slow');
Also of note... this IE bug appears to only apply to the jQuery show() and hide() functions. When I switch the animation to slideUp() or slideDown() the transparency works fine without any additional javascript.
So bottom line... there is something weird going in with CSS transparency in IE when applied to elements animated through jQuery show() and hide()
If I find out anything else in my research I will post it here.
Here is a link to my updated example, which works in IE 7/8 now.