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What was the result you were expecting?

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.

If the old URLs are being 301 re-directed to the new URLs, you should not lose much link authority, so that by itself shouldn't be a reason to change back.

If you prefer the two-level structure (which I would, for a couple of reasons) then go ahead and change back, but as Andy says, do it carefully so you are not sending conflicting messages.

When I try that page in the on-page grader, I get "Sorry, but that URL is inaccessible. " If I put that URL into Screaming Frog, I see it is 403 (forbidden). I can see the page when I enter it into a browser, however.

It seems that something is blocking robot access to this page (although I do not see a robots exclusion)...

Yes, as Patrick said, surprisingly often something like this is a result of a simple oversight because we have been looking at the same code over and over...

Do you have access to Screaming Frog? You could crawl your site and see whether redirects/canonicals are behaving as you expected.

Have you taken a look at the html of one of the incorrectly indexed pages when it is loaded in your browser? Can you see the canonical? If you try going to a redirected page, does it redirect? [I know--way to obvious, but sometimes it is good to start at the beginning again when we can't root out an issue.]

Another culprit in these cases can be internal links. Do you link internally using any of the undesirable URLs? That can send a message to Google that those URLs are still in play. Again, you can use Screaming Frog to find those strings.

To add to Patrick's great response, this page describes what you are seeing. When you try to go to a page on your site that does not exist, you seem to correctly get an error page.

But the content of the page does not necessarily reflect the HTTP response returned by the server. Your server is saying that the page was found, although the content of the page says it wasn't found. [A 404 response from the server simply means Not Found.]

If you look in your Google Webmaster Tools under Crawl > Crawl Errors you will see a tab called Soft 404. A soft 404 is a page that says it is not found but is not returning an actual 404, not found, response.

[I always wondered how Google could tell--I was given the answer that if Google sees that a whole bunch of different URLs bring up the same page, which does not return a 404 response, it assumes that that page says it is an error page and labels it a soft 404.]

If you want to test whether a URL on your site is correctly returning a 404, you can use Fetch as Google to see if it returns Not Found or else there are a number of free server response checkers online.

I haven't thought about it a lot because my issue resolves itself when I reload.

Have you tried entering the URLs in question directly into Open Site Explorer? What do you see there?

This might have nothing to do with your particular problem, but I have seen a similar thing on my site. When I see that, I reload the page and the Domain authority populates.

We never know exactly why Google makes the determinations it does. Maybe your competitor's site is better optimized for luxury cars, or maybe it has to do with links to the page. SEO is always done in a bit of a black box...

Google tries to show the result that is most specifically relevant to your search. When you search for "Exotic Car Rental Miami", "Exotic Car Rental Miami" is the most relevant part of the title. [Google uses the title and description you supply as strong suggestions, but it does not always use them.] Since you were not looking for Luxury Car Rentals, Google figures that part is not relevant to you.

I am not sure which parts of the rest you didn't understand?

When you do a search, you can click on the little green triangle next to the URL in the search results to see what the page looked like the last time Google crawled it, and the date on which Google crawled it.

As far as anonymous search, Google learns from your searches, so the results you see can be different depending on what you have done in the past. If you use an anonymous browser window (like Incognito on Chrome) you can avoid this.

Yes, for the most part "index, follow" is what you want on your pages.

But no, the "noodp, noydir" tags do not prevent you from being in the directories (though as Alan pointed out, the Yahoo one isn't around anymore), they just prevent the descriptions from being used.

Google does not always use the title and description found on your page, it sometimes chooses something it deems more relevant. Sometimes this is the description from the Open Directory Project (DMOZ). Sometimes this is not a good thing to choose.

Maybe your site was quite different when it was submitted to the directory, and the information there no longer applies. You want to tell Google not to use what is in there, so you use noodp in the header.

Whether you use or do not use "noodp, noydir", it won't hurt your rankings.

The only reason you might want to make sure to use them is if you saw unexpected content in the descriptions of your pages in the search results pages, and you had reason to believe the descriptions came from DMOZ. In that case, to prevent that from happening, you would use those additional tags, along with "index, follow".

When I look at your profile page, both OSE and the Moz toolbar tells me that PA=43. [Maybe you were just seeing a glitch...]

Look at the bright side, at least it's an easy fix.

Not at all, Dirk. I was just clarifying my answer.

If the pages still exist (just not listed on the website) and were doing well = Redirect or canonicalize to take advantage of their residual authority.

If the pages no longer exist = Not much you can do...

In either case, no problem with duplicate content as Google will soon figure out where the content now (exclusively) lives.

If you don't want to do the redirects, you can do the cross-domain canonical. Lots of unrelated sites do this, for instance when syndicating content.

I saw that the writer had said that site 1 no longer sold product range A, but I wasn't sure whether that meant that the range pages had been removed from Google's index or whether they were just no longer available on the site.

I also wasn't sure which site was the one ranking for great keywords. If it is the product range A products on site 1 (with the really good content) then it might be best to leave them indexed, with the redirect, till Google picks up on the change and passes the goodness to site 2. (If not, no harm done.)

Noodp= No Open Directory Project
Noydir= No Yahoo Directory

These are used if your website is listed in one of these directories with information you do not want used in the results pages. This might be the case if you have old, outdated listings that no longer apply. They tell robots not to use information from these sources, and they are optional.

Is your site prestigeluxuryrentals.com (from your sig)?
If I search on that URL, I see:
Title: Exotic Car Rental Miami
Description: Prestige Luxury Rentals offers the newest exotic & luxury cars for rent. Book an exotic car rental or luxury car rental in Miami, Miami Beach, and South.

If I click to see the cached version (it was cached on May 12) and look at the source, I see the same thing, with a slightly longer title, Exotic Car Rental Miami - Luxury Car Rental Miami Florida.

It wouldn't be odd for Google to shorten the title in the case of my search on the URL. (When I search for the full title, I see the full title in the SERP.)

Did you do an anonymous search when you were checking? You might be picking something up from your own history...

It is a surprisingly common thing to have happen...

You have it set to noindex.

<title>Automated Production - DNV Fabrication Lafayette, LA</title>