Chrome's speed advantage largely comes from its great Javascript engine. Since javascript is used on many websites and especially sophisticated web-apps like Google Docs and Flickr, many people will see a speed increase over Firefox.
Wait until Firefox 3.1 becomes available. It will have a new Javascript engine called TraceMonkey. The Firefox developers are claiming that it will be faster than Chrome.
I've currently got Chrome, Safari 4 (Beta), IE 8, Opera 9.64 and Firefox 3.1 beta 3 on my computer here for development purposes. Out of the IE alternatives, Firefox 3.1 subjectively seems the slowest, Safari 4 is a bit unstable, and Chrome generates user-specific identifiers and sends them to Google (there's a version called
SRWare Iron that has this patched out).
Firefox has an outdated codebase that is increasingly difficult to maintain. It's made up for somewhat by the enthusiasm of third-party plugin developers, but as far as the core is concerned, in many ways even IE8 is probably technologically more advanced nowadays. I'm mostly browsing with Opera.
Regarding IE 8 on unlicensed Windows: the problem is not the licensing, it's the product activation. It depends on how your particular cracked copy of Windows has been activated. If it's installed using an activation-less volume license key, it's unlikely that much is going to happen. If the activation has been disabled with a crack, IE8 may not work. In general, if you can run Windows Update without problems, you should be able to install IE8. If you can't run Windows Update, in all probability your computer already is a worm-ridden, remote-controlled, spam-sending zombie liability that should be taken off the Internet.
All of those are better than IE 6 or 7, though.