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As to #1, set your preferred domain in webmaster tools to either the WWW or without WWW version. Then set 301 redirects from the other to your preferred. This way you won't run into any issues of the bots seeing them as two different pages with duplicate content issues when they're really the same page.

Have you double-checked your rankings either through another tool or through unpersonalized searches to see if the Moz data is correct? I have a few search terms on two of my campaigns that regularly swing from 1st one week to "Not in top 50" the next according to Moz when in reality they barely shift ever.

Google changes title tags when a page is relevant for a query but the title doesn't appear to be relevant enough or, in some cases, is too long. There is nothing you can do about it except to possibly re-write your title to be more concise and user-friendly, and then hope that Google decides not to change it in other search queries.

But i've found that Google changing titles can be more beneficial than harmful. i.e. You rank better or look better in SERPs for related terms that you had not originally planned on. You can't stop all instances of Google changing your title, nor can you be certain of knowing all the search terms in which they have decided to alter the appearance of your title.

Alternately you could place canonicals across the newly bought site to the most relevant pages on your site and put up a message stating that Website X is now Website Y (with a link to your site) so everyone should change their bookmarks and start going to the other site. After a couple months, switch the canonicals to 301 to those pages. That way you don't wind up with bounces from people who wanted Website X but were confused/annoyed/perplexed/etc. from immediately going to a site they didn't want (even if its the site they need).

This makes me want to tell the story of Maille Ohye of Google talking about Canonicals at SMX East 2012. (I really like canonicals and probably talk about them too much on here)

She gave a hypothetical about having a fitness blog and taking over Matt Cutts fitness blog. Where canonicals came in was that she stated one of the primary uses for a cross-domain canonical was in passing equity from one site to the other site that will eventually be replacing it without causing confusion for users. So in the hypothetical she took over Matt Cutts' fitness blog, placed canonicals from his pages to their relevant counterpart on her site, moved content over as necessary, and placed a notice on the homepage of the Matt Cutts blog that it was going away but will be rolled into the Maille Ohye blog. This gives people enough time to switch over their bookmarks without bouncing from being redirected to somewhere they didn't expect. The canonicals would move equity to the main site, eventually pages would begin swapping out in the index, and eventually you would 301 redirect everything to their relevant counterparts on the main site once enough time has passed.

Following things that way I would assume that should be considered completely White Hat.

There are any number of reasons this could be happening. Perhaps they have more high authority links or maybe your site has a lot of links that don't pass much value. Their site could be older and getting a slight boost from that. They don't have social media accounts but they could still have been shared more on social media than your site. The level of traffic, bounce rate and exit % could be better than your site and Google could be seeing them as more relevant for those target terms because of it. Even though they don't update as often, their content could be considered more authoritative. Considering we don't know everything that makes up the ranking algorithm (& its constantly changing) and only really have concepts of the correlation between things, there's no way to definitively state that Tactic X is always the reason one site is higher in rankings for a specific search term than another site.

This is a common problem in WordPress. The tag archives, category pages and author archives can create many multiples of the same content and cause numerous duplicate content issues. Most people choose to NoIndex their archives since they have potentially little Search value but they do/can have value from a User perspective. Setting posts to show snippets on Archive pages can lessen some duplication issues as well. You should also double-check the tags you are using as one-off tags can cause a lot of problems with duplication. If a post has 7 tags on it and those 7 tags are only being used for that one post, you now have about 10 versions of that post sitting on your site. Make sure to only use relevant tags and that the tags used are then also used as needed on future posts in order to tie together related posts. All the functionality to NoIndex archives and change posts to snippets can be found either as part of your current theme or the Yoast plugin is a good download to help manage those things.

What sort of cards are we talking about? I immediately think "greeting cards" when you say that but I don't want to just assume that's the case. But if it is then from a personal user experience standpoint I would say that I would be more likely to search a specific range & gain more from finding a category page for a range of cards in the SERPs than I would from a page with an individual card on it. I.E. I'm more likely to search "birthday cards" or "get well cards" or "thank you cards" than I would to search "that birthday card with a grumpy cat that has balloons and a smushed cake". In which case I'd say go with robust category pages and, if possible, consider canonicalizing the individual cards to the category if you're going to also use the content of the parent category on the individual pages. If it's not greeting cards... well then I wrote all of this for nothing. (unless its playing cards... what I wrote might work for that as well)

As Logan said, you'd be better served handling these with 301 redirects. But you will also want to go in Google Search Console/Webmaster tools into Site Settings and set your preferred domain to either WWW on Non-WWW (depending on which you prefer to show across your site).

NoIndex won't cut the links. It will just remove the page from the SERPs. So you'll still be hit with the bad links to your site and organic traffic will be cut off.

Tag pages, category pages, and the like often have issues like this. One of the most common ways I've seen of handling Tag page duplicates is to NoIndex them. While they serve a purpose for the User as a means of finding related information, tag pages tend to be of far less use as an entrance page and can create duplication errors in the form of duplicate title tag, duplicate h1s, and duplicate content depending on how truncated the post is on that page. You will also want to look at adding rel="next" and rel="prev" meta tags to handle pagination (if you haven't already).

So every page has several hundred links to spammy duplicate pages? This is a bad thing.

First off, link equity leaving every page on your site would be split between all these several hundred links whether your NoFollow them or not.

Second, the fact that they are essentially duplicate of each other and there are potentially hundreds of these pages means that your site is likely hurting from thin content penalties.

Third, that image makes it look like old school 90s keyword stuffing. Which is just all sorts of wrong.

My assessment is that Yes... this is hurting your site.

In general, redirects can cause you to lose a small amount of equity. Both a 301 and a Canonical are forms of redirects except a canonical really only redirects the bots without redirecting the user. So much in the same way that it is not best to 301 to a 301 (though you can), it isn't best to Canonical to a 301. Now, it most likely won't hurt you doing that, if you prefer the easier of the two solutions, but its also important to remember that a canonical is a suggestion (not a directive). So the search engines can choose to disregard your canonical if they feel it is manipulative or not relevant. With this in mind, it is always better to point your canonical at the best, most relevant canon page instead of to another redirect.

You might be better served by using a canonical to point the parameters to the base page. I.E. /shoes?gender=1 with a rel="canonical" pointing at "/shoes". Depends on the variety of the content of the pages, if you're cannibalizing your own keywords, etc.

Personally I think you should set up a process whereby every time a vehicle and/or part is removed, you have someone automatically 301 it to the previous step in the site navigation. So when "blue widget 3" is removed from the site, anyone landing on that page or who has it bookmarked winds up on the "Widget" category page. Now there may not be an easy way to do it right this second because of how many there are now, but if you get in the habit of doing it and slowly work toward fixing the others then you'll be in a good position in the future to keep this from being an issue again.

Now if you really don't want to attempt that... 404s aren't necessarily horrible (too many can be). If your site is properly serving 404s then you won't be penalized for it but in this case you might want to consider using 410 status codes. Its a stronger signal for removal than a 404 and you don't plan on the product ever coming back so marking it Gone should get it removed from the index faster while also helping to keep you from competing against yourself in the SERPs when a new but similar product comes into stock.

Its important to remember that a Canonical is a suggestion not a directive. So making those canonicals is the right way to do it but the search engines and crawlers will determine at their leisure whether they believe the canonical is right or not. Sometimes its a quick fix, and some times they don't accept the canonical at all. If the pages are exact duplicates then Google will get the picture eventually and start recognizing it. As for the Moz crawler, I haven't had a paid account in a while so I can't speak to the way the system works currently but I do know that it used to be an issue where the Moz crawler would never seem to properly recognize a canonicalized duplicate... so that may still be the issue,

Normally I would NoIndex and/or disallow in Robots.txt a page like that.

Adding Nofollow to a handful of links on your site will not magically sculpt link equity in such a way as to create a noticeable improvement like that. If anything, you could just use robots.txt to remove those pages from being crawled. The bots don't necessarily need to index your login page, your human sitemap (if they already have their own), policies (which can change and cause legal issues if an older version is cached), and a few others.

And just a few months ago Gary Illyes stated that there's no good reason to nofollow internal links:

http://www.thesempost.com/google-dont-ever-nofollow-your-own-internal-links/

https://www.youtube.com/watch?v=GVKcMU7YNOQ

If a URL was indexed and has since had a canonical added to it pointing to another page, it will eventually disappear from results. Basically the pages gets consolidated with its canon page. If the bots choose to respect the canonical tag in that instance, all signals get passed to the canon page while still allowing the page and information to be accessible by human visitors. As such, there's no reason to keep the page in the index because you're telling the bots that another page is the correct page instead. This is not the same as NoIndexing a page but will eventually remove a page from the index much in the same way that a 301 will pass equity along to another page while eventually removing the redirected page from the index in favor of the page being redirected to.

Not exactly. Its not so much that the canonical "supersedes" an index, follow tag.... a canonical tag establishes equivalency while a NoIndex is more like a "does not equal." The Index, Follow is still there and being seen by bots as they crawl... in fact, if you had NoIndex on a page with a Canonical Tag, it may not even see the canonical at all since you told it to NoIndex the page. The Meta Robots Index tag comes first allowing the bots to crawl and index the page but then the canonical sets up equivalency to a separate page. So if your canonical tag is being respected, it doesn't wind up doing the same thing as a NoIndex (though it may seem that way) nor does it do the same thing as a 301 (though there are similarities in how equity is passed). Since a canonical establishes an equivalency, you'll find that the Canon Page will eventually take the place of the Canonicalized Page in search results because you're telling them the Canonicalized Page _is _the Canon Page & that the Canon page is the right version of both.