I just checked that link in Screaming Frog and it is 200.

My redirects from old domains show up as back links, which makes sense--it is a case of a link from one domain to a different domain. So if you redirected all of those pages to your new site, you'd see a lot of links. As far as why it would be more pages than you had, take a look at the links and see what they are. Maybe you had different versions of pages that are showing up as separate links?

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.)

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.

You should not look at Moz's domain authority as a fixed number; it will change, depending on their index that month. A good way to look at this number is relative to a couple of your competitors. Track them, and if their number also goes down when yours goes down, it is probably the index.

When I go to your site and search on USB flash drives, I see 53 products, same as in the Google snippet. What number did you expect to see? http://imgur.com/ZKZGvyb

[I know you obscured the domain in your image, but the name of the company is in the title.]

Why do you have it set up that way to begin with? Could you put the content on the higher level page? Or is it meant to be an easy-to-remember "vanity" URL? In any case, yes, do not put a page with no content in your sitemap.

And to get it to stop showing up in search, you can add a canonical to the page, pointing to the deeper page with the actual content.