Thank you so much for all your tips and answers. I was forced to move on today due to a deadline and switch to jQuery, but this will for sure not be the last time, that I'll be needing similar functionality, and will definitely try this out next time. It's also good to know, that you guys are fast and thorough when responding to any questions, that might arise while using Greensock for the first few times.
"1: you were setting opacity:0 in your css for .tooltip. IE8 doesn't recognize opacity so that is why all the tooltips were displaying initially and the tweens didn't work."
In the actual tooltip menu, that I was working on, there is of course a lot more styling on the tooltips. They have a border-radius in all corners and box-shadow, and I'm adding support for those in older IE browsers by using CSS Pie. I was convinced, that the problems I was having, had something to do with the pie.htc. For setting opacity for older IE I was using the required alpha filters, but you're saying, that "visibility: hidden" should be used instead? I'll have another go at it later on, when I'm at the office again.