You must also remember that if you were using any MS office product such as Word anything that you have cut or copied during your session is left in the clipboard until you delete it. Which means any of that info is also accessible via websites if your online at the same time.
Your clipboard can hold a lot of various info that you may not even recall putting there

This clipboard "feature" of IE is something I had mentioned in a browser debate concerning security a few years back and no one believed me, now that MS themselves have made it known to the public people are pissed about it.
As for websites "requiring" you to use IE that is easy to explain it is due to web design in general, the website was designed using IE most likely and won't display in other browsers that follow the standards set by the W3C. The W3C writes the web design "law" but unfortunately MS feels less than compelled to follow a set of standards designed for the benefit of all who use the web by allowing web designers to design pages that can be viewed by any browser and still appear the same without having to do a lot of hacking. This is a major headache for people like me designing sites, trying to make a site that will appear the same in all browsers.
Then there is ActiveX controls that many sites will use that other browsers won't use due to the security issues with it. Thankfully my bank has realized the problem and no longer uses stuff like that. So i can safely bank online using other browsers and not be tied directly to IE.
There are vulnerabilities in all browsers and there always will be as long as there are those searching for ways to beat security measures. It is just that some browsers are designed with less "features" that actually create those vulnerabilities.