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You can use a cross-domain canonical from site 1 pointing to to site 2, or 301 redirect the pages from site 1 to site 2.

Duplicate content isn't a penalty, it just makes Google choose which version to show. If you use one of those signals (probably the 301, if you are sure this is a permanent change), the correct site will get the benefit of the content.

Put a canonical tag on site.com/products to itself and the parameter versions will go away. (They eventually will anyway--duplicate content does not cause a penalty, it just causes duplicate versions to not be indexed, which is what you want anyway.)

And I forgot to specifically say that I would definitely choose to update these if possible, rather than delete them, because they would also be strong signals for your current company name.

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.

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.

There is some indication that Google may discount content hidden in tabs or click to expand, even if it is genuine content. (John Mueller of Google said this in a Google + hangout--see this article by Barry Schwartz on the topic.)

A way to get a quick result is with an Adwords campaign. Create a landing page for people who are looking for the website that is gone, and explain how your website is filling that niche. In the meantime, keep building your organic presence. [Easier now, without the big competition.]  After people have learned to come to your website, you can back off the paid search and refine your organic approach.

Yes, that is what is happening. Google will often use the title that is on the page, but it does not always do that. It will use a different title if it thinks a different title is more relevant to the searcher's intent. This is especially true if the content on the page does not match the title. (For example, on the Elexio page, the words "Church Management Software" are not even present on the page, other than the title, description, and schema.)

As far as a reference for the answer that the keyword should be near the start of the title, Moz's Beginner's Guide refers to this.

But what I find interesting is that if you look at those pages (not just the serp), only two of them actually start with the company name in the title on the page. The others are just displayed that way.

I just came across this article, which sums it up.

The site will tell you when you go to submit your site. Here are some examples from JoeAnt and DMOZ.

Moz used to have such a list, but found keeping it up was beyond the time and resources they had available. The general advice given about getting listed in directories is that they should be good quality curated directories, relevant to your business.

In general, directory links are not as valuable as they used to be, and there are many low-quality ones out there, so be careful with this type of linking.

I see the same thing as Jonathon. Though a whois search says it doesn't expire till this August (and has been registered since August 2009).

Yes, you should be able to do that as well. If you can't, that is a problem with that tool. Maybe someone from Moz could take a look...

Where do you mean, in one of your campaigns on the on-page optimization page? Moz will grade any page that has a keyword ranking in the top 50. So if you see a keyword associated with a page there, it is because that page ranked for that keyword, whether or not you had intended that.

It is hard to answer without seeing the pages in question, but there are ranking factors other than on-page that affect position. Maybe page B was less optimized but had a better backlink profile, for example.

It is always a good idea to work closely with your developer when making changes like this. I'm glad you got everything figured out!

If search engines did not catch up with changes we make and improve our ranking for positive changes, there'd be little point to Search Engine Optimization.

If Google is already seeing your pages anyway and the move will only make them better (even if they are still not where you'd like them to be), then you can go ahead and move them if you like, as long as the move will not create a confusing situation for the people looking at the pages.

As you fix the pages to your satisfaction, wait for them to be crawled again or resubmit them using Fetch as Google to possibly get them crawled faster. [And as far as H2 tags, if that is your main worry, I wouldn't worry too much--they probably won't make much difference.]

I sincerely hope not, as I am about to redirect most of my main website.

It is a common thing for websites to do a wholesale redirect of pages, whether to change domains or better structure the site. Matt Cutts said a few years back that there is no problem with this, even with sites of 100,000 pages and I have not heard anything to contradict this.

Why are you switching? If there is no reason to be in a rush, then I'd wait and make the change when everything is ready--a few weeks isn't that long.

If there is a particular reason for haste (like you were having technical problems with the old platform or a lot of your traffic is mobile and you want to make the April 21 Google deadline), then I think it depends on the state of the content.

If it is not perfect but still makes sense with the new titles and URLs, I'd do the update for your most important content and switch. If it is terrible, I'd wait. There is no point getting traffic for bad content.