It depends on the format. A plain text http://mysite.com will not be counted as a backlink, it must be in an anchor tag like http://mysite.com and the link must be followed, ie not include 'rel="nofollow"'.

My understanding is that the same URL is used for both test and control versions of your home page, meaning its not actually redirecting. Instead the changes are inserted using the javascript code dynamically. I'm not sure whether Google ignores the javascript change or incorporates it. Either way your test changes should be done with both SEO and user experience in mind, so don't run tests that would hurt your SEO and you'll be golden.

If its possible to fix the site without altering the directory and URL structure, that's going to be your best option.

However, I'm guessing you're going to be fixing URLs as well, and if that's the case you'll want to use 301 redirects to tell search engines where the content has gone. See http://www.tamingthebeast.net/articles3/spiders-301-redirect.htm

Finally, wherever possible try to get the owners of the sites with backlinks to update their links to you.

Have you already tried www.xml-sitemaps.com ? It will crawl and generate a sitemap in whatever format you want.

However, in terms of reporting problems I've found the SEOmoz crawler and related tools to be invaluable.

You can save your sitemap in xml format from the above site, and use that to submit to search engines.

Woah, quicker on the draw! I yield to you, sir