Can you give some examples of these 'favorite websites'? I'm interested in knowing what kind of website would be so interesting that I would want an entire offline copy of it. (Besides maybe Wikipedia)
Mostly defunct webcomics but also some of the small personal sites that documented and collected resources for particular events or strange people. A lot of those arose out of the SomethingAwful forums. For example, there was a guy named Brian who wrote batshit insane fanfiction about himself. One of these sites archived the fiction, interviews with Brian, videos, recordings of collaborative reading skype parties, etc, all neatly on one site and now safely tucked away on my drives. Now I have a little piece of nostalgia from 2004 I can step back into.
Are you able to navigate through the sites using the original links? I notice that on the Wayback Machine, internal site links only work if that particular page was also archived.
Yes. It also lets you designate a particular number of "steps" outside the original site it will also archive. So I give it a site and a 1-step limit, I'll get the site and also any individual other webpage linked to somewhere on the original site. It doesn't do so well with modern sites that full of CDN-hosted content and pages that depend on data from two dozen different domains to function properly but it's great for old pre-web2.0 stuff.
The Rotten Library is the first thing that comes to mind.