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Along with Patrick's great ideas, take a look at Wikipedia and DMOZ (Open Directory Project). Is the old company listed in either of those? Sometimes signals from those sites can be very strong.

Google doesn't guarantee that the author picture will show up and whether or not the picture shows up varies, depending on the searcher's search term.

Writing more under a particular byline seems to help.

Judging by the screenshot, these are 4xx errors from your Moz crawl, yes? If you look on that spreadsheet in column AM, "Referrer" it will show you the referrer of those URLs. Look at those pages to see where the errors came from.

My guess is that these are being generated by your own website. This type of thing can happen if there is some confusion with absolute and relative URLs and the wrong form is being input somewhere.

Could it have something to do with your topic, rather than the site itself? When I look at "student debt relief" in Google trends, I see a big spike in July, and September is half that. I know on my site that traffic is related to seasonality.

Hopefully someone in SEOMoz technical support will notice this question and comment...

You request an update in DMOZ, explaining about the change. They are much quicker at updating an entry than at adding a new one. (I know, because I had to do an update myself.) There is also a handy meta tag you can use on your site, , which tells robots not to use Open Directory information.

For Wikipedia, if you know a friendly editor you can ask them to make the change, or else you can request an edit on the company talk page.

G+ results have been seen in the results pages on page 1, though I can't say I've seen much of this. Google+ Profile PageRank: The Real AuthorRank? - SMX Advanced 2013 

Moz and SearchMetrics have both reported that a URL's number of Google +1s is very highly correlated with search rankings. Matt Cutts has said Google doesn't use that as a ranking factor.  (And correlation doesn't equal causation, as we know.)

I don't know of a good metric (though I'd be very interested in hearing if anyone does) but at least for now, I consider Google+ an important factor.

I started to answer this one way, but then I looked again and you were looking for links from your website to external sites that are broken, yes? I do not know of a way to do this on Moz (though you can look at broken internal links), but you can do this with Screaming Frog. (A great resource to have for a lot of things...)

Actually, someone asked a very similar question to this just around the same time as you did. The responses in that thread might be useful to you: http://moz.com/community/q/how-to-use-robots-txt-to-block-areas-on-page

There is also information on internal links in Google Webmaster Tools; under "Search Traffic" in the left column, choose "Internal Links."

If you set up a ppc campaign for something you'd like to rank for organically, you can get information you could not get from Google otherwise, such as what searches are bringing up your ad, which can help your organic efforts going forward. It is also a way to get people to see your information when they may not have been able to find it organically yet. It sounds like you have the money available to test this, so try it and see what results you get.

There is some question about the fate of Google Authorship; in fact AJ Kohn says that the Authorship Project at Google has been abandoned and Google will be using other ways to extract identity from pages.

http://www.blindfiveyearold.com/authorship-is-dead-long-live-authorship

On that spreadsheet, in column A you will see the URL with the redirect. Column G is "Temporary Redirect" and those should be TRUE (on the 302 redirect report). Column AK is Redirect URL and it is there that you will see the page that the page in column A is redirected to.

Are you looking for something other than this?

Yes, being indexed first is good, but then again a higher authority site tends to be crawled more frequently so even if you post first, you might not get indexed first.

Going back to your original question, if the higher-ranked sites that also sell your product choose to keep their canonical pointing to their own site, they may well outrank you. It sounds like having your listings on other sites is part of your business model so you can't do anything about that, but maybe you could create some content that is exclusive to your own site and have that rank.

Also, if you have a number of higher-authority sites that use your content and have canonicals pointing to you, that would be a clue to Google that yours is the original content, even if one of the sites keeps the canonical pointing to itself.

In general, you will get the most links to a page via whatever URL is easiest for the linker to grab, so often it is not the prettiest, with parameters and such. But you really don't want that showing up in the serps, so don't use that as a canonical.

As far as the trailing slash/no slash issue, most people will use the no slash version if they are choosing how to add a link, even when the slash version would be more correct (as in your example) so you would also expect to see more links there.

But you mention that the slash version is the ranking version in your example. Is this mostly true throughout the site? (Maybe it is a Wordpress site that ends everything with slashes?) Then I'd stick with that. (I myself use the rule you first mentioned, the cleanest, non-slash version, but my site doesn't use trailing forward slashes.)

Do keep it consistent though. It would be a pain to always be checking pages to see what the canonical should be. The amount of SEO value you might lose is minimal.

How is the name written in other types of official correspondence—letterhead, signage, etc? The look of the name should be standardized across all sources, but that's a bigger issue than just SEO.

In terms of search, using "and" or "&" in searches will likely give you different (but very similar) results—try searching your company name with each one and see what Google already thinks of the two variants.

Using symbols like "&" can be more eye catching than just letters so that can be a benefit on a crowded search results page, depending on what else is there.

Absolutely; you should implement it anyway. Even if Authorship doesn't turn out to be as important as it was first thought it would be, it is another signal to Google.

That is the only spam flag I get as well. The explanation is that: "Non-spam sites tend to invest in rich user experiences with CSS, Javascript and extensive mark-up. Accordingly, a large ratio of text to mark-up is a spam signal." I am not too worried about it--a fast site makes for a very good user experience, I think.

You started by saying the problem is duplicate content. Are those pages with the various parameter strings basically duplicate content? Because if they are, no matter what you do you will probably not get them all to rank; the URL is not your main problem in that case. (Though you still should do something about those parameter strings.)