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Google is constantly crawling the web. When I search "next google index" in Google in quotes, this post is the second result on Google right now. They do tend to crawl different sites at different rates, however. They crawl CNN a lot more often than they crawl the local RC airplane site that updates every three months, for example.
My husband is here having a good laugh with me over the URL that I brought up just as he came in from the garage. "It's from Q&A and it's work related honey, honest!" 
Have you viewed your source? Do you know about the link to an adult site with an H1 about bondage that's in there? Your privacy policy also has a link to car insurance on it. There's an affiliate link to the Socrates theme, too. If none of those are yours, I'd suggest removing them. Might help, certainly can't hurt.
I put one phrase from your site into Google, and found a couple of dozen sites that had that phrase. To be honest, you've got a real thin site without much content, and if the content you do have is not unique, there's not much compelling reason for the search engines to return your site instead of another site. Open Site Explorer reports links from nine different domains, and most of them look like profile links.
I'd first focus on getting some unique content in there, making sure that your site is clean without hidden links or text, get content in there that someone would want to link to, then focus on linking.
What you're seeing may be related to last week's Linkscape update. Rand has a blog post about it over at http://www.seomoz.org/blog/linkscape-september-update-feedback. Take a look at it and see if that answers your questions, and post any follow up you have here. Thanks!
Right now we only integrate with Google Analytics. We do have a feature request forum, where you can add your vote for Piwik integration. There's already a thread started at https://seomoz.zendesk.com/entries/20870741-Piwik-Integration.
It could be some of each. I know it's frustrating when you don't see those links that you know are there, but the good news is that Google does see them. The good news is that we're getting indexes out there faster than we ever did before, so it's every 1-2 weeks these days instead of 4-6 weeks.
You might also want to review the section of the beginner's guide to SEO that talks about links at http://www.seomoz.org/beginners-guide-to-seo/growing-popularity-and-links, and the link building section of our blog at http://www.seomoz.org/blog/category/4.
Google trends has this. Check out http://www.google.com/trends/?q=unpasteurized+milk.
If the page is not crawled or indexed by Google, then presumably they haven't seen the links and it doesn't count in the algorithm. It can certainly count in the minds of users who view the page. If I had a link to my site on a password restricted website that was used by the top writers at TechCrunch, I'd be quite excited because real humans are going to see that link, visit my page, and possibly write about me.
If a low-quality website has made a page that is titled "spammy link exchange page that I've noindexed and going to see how many suckers I can get to give me links that I'm really not reciprocating on" then no, that's not going to count for anything.
There is not a way to change to Google, as they've no longer made that data available, unfortunately.
When you put your own URL into Open Site Explorer, we're looking at that URL (such as mycompany.com) and the FB likes that URL has.
What you've given me here is the URL for your Facebook page itself, which is actually a completely different site.
Hope that explains things!
Aside from the domain name, is the URL structure identical between the two sites? If so, there should be a real easy redirect to just say "if person hit oldsite.com/page123 take them to newsite.com/page123". I don't know that exact code, but can get someone on here who does.
It's a small online store -- how are conversions and sales doing? Traffic and rankings are great, but if I (with an online store that sells items related to RC warships and scale boats) somehow rank for rabbit cages and get a lot of traffic from it, I've got rankings and traffic, but it hasn't done me a bit of good for the bottom line. If you then come along and get me three link related to DKM Bismarck plans that have 1/10 the traffic I got from the rabbit cages traffic, but that traffic converts and gets me sales, I'm going to be happy.
tl;dr Are you and the customer focusing on measuring the right results?
The upgrade hadn't yet happened when you posted this. I'm not aware of a criteria change, but will ask to see what could have caused this.
You should start to see some changes in the next OSE update.
Hi Dana,
I don't know the operation of the toolbar well enough to say, but we did roll out an index update this afternoon. I think it started after you wrote this, but for the heck of it, could you come back and look at this and see if you're still seeing the same PA on every page you visit? If so, it's probably something to send an email to help@seomoz.org about, as they can look into it better.
Thanks!
That explains US and Canada, but the OP is actually asking about the UK. I'm not aware of a partnership there.
Edit: there is an agreement for the UK as well. Details are at http://www.mediaweek.co.uk/news/1083306/Yahoo-UK-searches-switch-Bing-results/.
Just wanted to leave a quick note saying that SEOmoz has upgraded the Link Directory! You can view a post Cyrus wrote with more information about the directory updates and link building via directories at http://www.seomoz.org/blog/seo-link-directory-best-practices
As a side note, we now have http://health.moz.com/ where we list trending issues like this, and in the case of the GA connection, the solution. We hope this will make it easier in the future when events like this happen.
A couple of notes here first.
Google has really clamped down on private blog networks. Your effort might be better spent in other areas than developing a network.
Domain Authority is a Moz metric, and the search engines don't use our metrics for rankings. We approximate, to a degree, how we think the search engines might value sites.
How the search engines treat an expired domain may depend on what you do with the site itself. If it had information about Ford Explorer tires and now it has information about hot air balloons, they can tell that something big has happened.