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What do you mean "pages that have nothing to do with my site" are these not on your domain or are they on your domain but you are not familiar with them?

If your duplicate pages are behind a log-in you will be fine as content behind it cannot be seen by search engines. You should also block your logged in pages using your robots.txt.

You should not lose historic data but re-linking the account with your credentials should fix any problems that you are having.

There are a few things you will want to check.

Make sure that the account that linked the GWT and GA has not changed (such as an agency account did it then left) because the owner of the GWT account determines the verification code.

See if they updated their Google Analytics to Universal, if they have the wrong code/combination it can cause problems.  Also with using the GA code to authenticate, make sure that it is placed in the right place (you can use Tag Assistant browser plug in from Google).

As for why it went away, Google will periodically check the authentication to make sure it is still there in case ownership changes, so it is not odd after site changes for this to occur.

I completely forgot that screamingfrog had a built in sitemap generator, good call.

Unfortunately the site I have does not automatically generate its sitemap.xml nor can I have anything installed onto the site to do it.  What are some good paid tools (As most free tools either have a limit or a term against business use) that I should look into.  Lastly, I do not want any type of subscription as I will have an automatically generated sitemap.xml with an upcoming site launch.

The answer for this is pretty simple, both tools are not the same nor do they look at the same data. While both tools are part of Google, they each interact with search differently and measure impressions in different ways (such as how image search is measured, map searches etc.) I do not try and draw direct comparison between the two tools, use them independently for their niche and you will be fine.

Here is the tool and some instructions from Moz on using it:

https://www.google.com/webmasters/tools/disavow-links-main

http://moz.com/blog/guide-to-googles-disavow-tool

With Yoast, it creates different site maps for pages, posts etc.  I have found that indexing the various sitemaps and not just relying off of submitting /sitemap.xml helps with indexing.

Yes, I have a lot of experience with GTM.  It is designed so that anyone can add marketing tags onto a site without needing to involve your developer.  It is a very powerful tool that you can use to define very specific circumstances with to consider conversions to measure external marketing such as banner ads etc. with.

They could be using it to track what keywords are being used, it is a poor choice as linking GA to Adwords is a better option, or using {keyword} to get the actual keyword used instead of hard coding in each and every url from Adwords. They might have a system in place that needs that, but from both a toll perspective and and SEO perspective there is no beneficial reason to do that in my opinion.

I think you are more spot on with your 2nd option, creating content worth linking to is the best way to get real, quality links.

You should disavow those links, if they are not effecting you now, they will eventually.

As for why you are not back to the top, you changed your site's focus so you lost all of the historic "mojo" that you had for those keywords.  Even now that you have put them back, what was lost is gone, you will need to slowly, overtime rebuild to get get back to what you had before.

If you cannot dedicate the time to SEO, bringing in a professional may be your best option.  They can help you get on the path to recovery and work with you to hopefully improve past where you started while fixing what this other company did.

Yes you can, make sure that you don't just add it into the button but you use an onclick command to call a script that will run and place the code onto the page.  Be careful because if you do it this way it will send a conversion on the button click even if it does not lead to the proper conversion (such as if you placed it on a form submit button but the user had not completely filled out the form).

Most likely they are if the situation you described is how they have gained their links.

Best practice you should put the canonical tag onto the main page as well as the other pages.

SEO is not all back links, content is king, traffic is important and on-page problems can drag you down.  Instead of worrying about rank and back links you should work on a content strategy, make sure what you control is optimized (fast page load, no huge errors, proper meta etc.) and work toward small, sustainable incremental gains.  If your competition is using bad tactics it will catch up with them, you can grow your position now and be primed to take over when Google catches them.

While they as a whole have high page authority, when you create a sub domain blog on them, it does not get the same.  You should worry about creating good content on your site instead of making free blogs to put content onto to then push to your site.  Your best bet would be to set up a blog onto your website so that you can have new, relevant content being consistently added to your site.

I would not try to sell this anymore, the times have changed and unless those directories are driving conversions I would avoid them, paid directories have no place in SEO according to Google.

Then the answer is no, do not worry about keyword/content optimizing pages that you do not want ranking.