What exactly is a "natural" link then? So you're saying that as a vendor if my manufacturer requests that I have a link back to their site that is an "unnatural" link? Links don't grown on trees. People request them, people place them there to give credit. They are used as footnotes, examples, sources, etc... The only unnatural link out there are the ones that the website owner did NOT place there intentionally. For instance, spam links, hacked links, or embedded links in blog/forum posts that are specifically placed to promote a product, website or whatever. In other words, "unnatural" are unsolicited links. A Natural Link is a link which a site owner clearly intended to place on their website. The reason, relevance or method has no bearing on the nature of the link.
Running? No. Does the site still exist? Yes. As far as I know all links were deleted. Either we're going to let it expire or change it to something else.
I agree that by adding a no-follow tag means that they declare: we didn't want ourself as a directory.
Real directories or link farms calling themselves directories? Name the directories listed giving you penalties.
As long as he's up front about it, then I don't see the problem. (Although I wouldn't personally use it as it's very unlikely you'll get any click through traffic from a listing)/