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On the campaigns page (the pro home) under the grey bar that says "Campaigns", there is a white bar that has 5 of 5 campaigns (or whatever number you have) on the left--all the way to the right on that bar, in light blue, it says "Manage Campaigns." Click that and you'll see where to delete (or archive).
Fix the tags and don't worry. As you say, Google will re-crawl and the issue will be resolved.
Dashes are Google's preferred word separators. (I have heard that underscores are considered to be word joiners, so in that case your first one would be carfuel rather than car fuel but I am not sure whether that is correct.)
In any case, the two would tend to look the same to your human readers and could cause some confusion there, so if you have the option to change one, I would.
Matt Cutts has said a number of times that invalid HTML in and of itself does not cause a penalty. Here's one example: http://youtu.be/j3KgrbiB1pc
"So Google does not penalize you if you have invalid HTML because there would be a huge number of webpages like that and some people know the rules and then decided to make things a little bit faster or to tweak things here there and so their pages don't validate and there are enough pages they don't validate that we said OK this would actually hurt search quality if we said only the pages that validate are allowed to rank or rank those a little bit higher."
Yes, you should write good HTML. But, unless it is so bad that it causes a bad user experience, simply having some W3C errors is not going to be a problem for SEO. (At least for now...)
When I went to look at that suspect site the page said it had been suspended, so you are certainly right about the suspect part. What I have seen sometimes when I start getting a lot of referral traffic from an unknown site is that the site is taking my content and putting it up on their site, links and all, presumably for the sake of ad space they are selling on their site.
I have a complex CMS for my main website; I just asked my developer to do it. On my Wordpress sites, I use an SEO plugin for this (Yoast).
If you have four different sites with basically the same content, meta descriptions are not your main problem. Google does not use what is in that field for ranking purposes; the purpose of writing good, unique descriptions is to entice people to click on them. Your big problem is that four extremely similar sites means Google will see them as duplicate content and one at most will end up ranking.
When I go to https://www.qjamba.com, I am not redirected to http://www.qjamba.com, so you should check on your redirects.[In fact, I seem to be 302 redirected back to https://www.qjamba.com!]
And once the redirects are correctly implemented it might take some time for Google to find them, but when that happens the https version will not be indexed.
To add, the other thing as far as point two, you mentioned that they checked Google on their own computers. Have you talked about how Google tailors search results to particular users? You could send them a Moz keyword ranking result, with personalization removed.
Google can and does index PDFs. Here is an article about pros and cons.
If you are adding value to the category pages and making them useful places for the user to land and not just a list of links, then by all means, you should keep them indexed. And yes, in this case you still need for each page to have a canonical URL, so you will have to pick which category a recipe best fits in. (You can still have the link on two category pages, one of them just needs to be the canonical.)
There are a number of ways, along with the source code. Open Site Explorer will tell you, as will the Mozbar. There are also browser plugins you can use.
If you want to look at your most recent crawl report, go to your campaign from "campaigns" on the Moz ribbon and click "view this campaign". That will take you to the overview page, where you can click "see all crawl alerts" which shows you your crawl report. You then have the option to export it as a pdf or csv.
Yes, it is indexed and shows up on the fourth page of results (#33, according to RankChecker). Sometimes a new page can bounce around in the rankings a bit.
The last Mozscape index update was 7/11/2013. In August, they put this up:
"Dreadfully sorry, but we've decided to not release our most recent Mozscape Index, since it was not up to our quality standards which we measure with index correlations.
Why did this happen?
Our scheduling algorithm always tries to add new URLs to the schedule to make sure we expand our coverage over time. Unfortunately it seems to have hit a new section of the internet that we never crawled before. This resulted in a huge set of new URLs that seem to be of low quality, and the this has skewed the rankings slightly (a little more than we are comfortable with). Large domains seemed to be most impacted and showed a marked decrease in external links.
What do we plan to do now?
We are attacking this from multiple angles:
When can you anticipate the next index release?
August 26, 2013"
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.
The on-page grader lists the pages that rank in the top fifty for a keyword in one of your campaigns, so it is not the on-page grader that is choosing a particular keyword for a page, it is Google. 
In other words, even though your "bracelets" page is not optimized for "rings", it still showed up for "rings" in the search results. And obviously since you did not optimize it for "rings" it did not do well for that word.
When you see a grade of "F" for a search term that doesn't apply to that page, you don't need to worry about it. Sometimes pages do rank for keywords other than the ones we are optimizing for. If you get a bad grade for a search term/webpage combination that are supposed to go together, that is the time to go back and re-optimize.
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.]
In your chrome browser, there is a little "hamburger menu" on the top right (three horizontal lines, stacked).
Click that, go to tools > extensions and you should be able to delete it there.
ps- What do you find that it covers up?
A webpage can rank for more than one keyword and that's a good thing. Having the low volume keyword ranking well wouldn't hurt the main keyword's ranking. However it could be that Google's algorithm decided that the page is better suited for the lower-volume query. I would take another look at the page, including links and citations and see if you see something there.
For the homepage issue, it still sounds like a subdomain issue, especially because you see it when you search the root domain, but not the www subdomain.