I have once run IE6 scripts on IE7, and found the caption of the pop-up window was different, which resulted in QTP unable to find the window. After replaced the window caption of IE6 with the one of IE7, the scripts ran well.
Also with IE7 you will have to disable the tabs, QTP will take care of this for the first run , but you have to restart QTP after that.
And in IE7 you will have an infomative page for the invalid site certificates, see the attached.
"I realize it's an error, but no one is going to try to do that!"
From "Top 10 Stupid Comments from Developers".