I see the opposite, Firefox is correct showing the text in the middle of the blue buttons but IE shows the text on the buttons too high, so I left all the coding as you had it for Firefox but added a conditional comment that only IE operates.
Put this AFTER the main script and styles:-
<!--[if ie]>
<style type="text/css">
.suckertreemenu ul li { margin-top: 10px; }
.suckertreemenu ul li ul li { margin-top: 0; }
</style>
<![endif]-->
You still have two <html> codes and two head sections, the main ones are after body markup so they ought to be moved up and consolidated. However, I left them as they were and as long as you put the conditional comment afterwards it will still work to lower the text in the buttons in IE. It's always best to get Firefox correct as this is usually OK for most other browsers and then you can put a conditional comment in just to edit IE.
It looks OK in my IE7, Firefox and Opera but Safari still shows the text too high in the buttons. Your code is still a bit of a mess and needs sorting out, possibly starting again.
Code downloaded to my PC will be deleted in due course.
WIN7; IE9, Firefox, Opera, Chrome and Safari for Windows; screen resolution usually 1366*768.
Also IE6 on W98 with 800*600 and IE8 on Vista 1440*900.