Are there any IDEs for Javascript /HTML that allow visualizing the effects (like MochiKit), and streamline the website design process?
I really like Visual Studio 2008 for HTML and JavaScript editing. With VS2008 SP1 you also gain JavaScript Intellisense to help you write code quickly.
I was a big fan of Nvu, which was Linspire's fork / update of the old Mozilla Composer. I'm a little startled to see that that project has apparently been abandoned and resumed by a different group as KompoZer. I haven't tried it since the name change, but the old Nvu was fantastic (also FOSS.)
Aptana Studio is a popular choice. It is based on the Eclipse framework, so is cross-platform (Java) and well supported; it seems everyone is building an IDE on top of Eclipse these days (Adobe Flex Builder, Zend Studio, ...). Aptana Studio comes in a pay-for "professional" flavour, or a cost-free edition, and there are several Aptana plug-ins you can add to support, e.g. Ruby on Rails development.
Eclipse has a built-in web-browser, but will also work nicely with any browser on your system, you just need to configure it. There is also a web development tools package, which includes a WYSIWYG HTML editor.
I'm actually not a fan of Aptana. I already use Eclipse PDT for PHP development and installing the Aptana plug-in messes with my preferences, so for JavaScript support I use JSEclipse (sorry for the lack of a link - Adobe have rolled the functionality into Flex Builder, so it's hard to find details of JSEclipse itself now). You can configure Eclipse to download and install JSEclipse from the JSEclipse update site.