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I'm not sure there is a way to segment it because it is Organic Traffic and I don't believe that there is any identifying characteristic in Analytics, although I would love for someone to find one. While you may not be able to glean data from Analytics, you can at least use the Google My Business Insights Visibility report to find out how often your local listings are showing and what kind of actions are being taken. It's not ideal, but something is better than nothing.
A general rule to follow would be that if they are offering a package deal, it's probably not good. What works in one niche may not work for another. A lot of their link types are considered spammy and they are offering a ton of items for a very low cost which throws a lot of red flags.
The most likely lines in your robots.txt to cause this:
User-agent: *
Disallow: /
These will likely clear when whatever tool you are using crawls them again and realizes the links aren't there anymore.
Sometimes this is done unintentionally, and there may be no ill-intent. Either way, one should be properly redirected to the other.
The sites you are checking may not be in the index for OSE, Majestic, or Ahrefs. These tools don't have the resources to crawl every site and page on the internet, but they are growing all the time. Don't let it not showing in OSE stop you from going after a good link.
A good reconsideration request does three things:
It would help if you post a link to the page Google sees. You can do a search on Google with the site: operator to find out of the link is indexed. Basically, you need to figure out if the link exists and then figure out what is causing it.
There are some base 64 encoded URLs on the page. They show in the source code like below. That would be my guess as to what is creating the links, which are obfuscated for users. These types of attacks are usually called in your functions.php file or within a hacked plugin, or could actually be inserted into the css as well.
background:url(data:image/png;base64,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)
I think either way is fine with #1 being better, but I would also add the name/brand of the store to this as well. Users wouldn't want 10 listings that said the same thing, they would want some way to differentiate.
A lot of things happened here in combination. The new site launched. You say a disavow file was updated, was there a penalty or was this pre-emptive? You might have killed a lot of links that were helping you with this. Looks like you changed CDNs. The site was switched to secure, except for the forum, and a lot of the secure pages are displaying non-secure content? It looks like you added more ads to the site. You removed some urls in Google? You should really wait for the pages to switch naturally through the 301s to the new structure. What you may have done is deleted pages that would be showing right now while the new pages haven't been crawled and added yet. There are a ton of broken and redirected internal links after the new launch. Some old pages were never redirected and I would be willing to bet there were some links lost. Looking at a quick speed test on pingdom compared to the history, it looks like the site is taking twice as long to load now as it was before. There are some ridiculous redirect chains happening that need to be cleaned up. A lot of pages are redirected 2-5 times before reaching their current destinations.
One example:
http://bellybelly.com.au/articles/Breastfeeding/breastfeeding-a-confidence-game
http://bellybelly.com.au/articles/Breastfeeding/breastfeeding-a-confidence-game/
http://bellybelly.com.au/Breastfeeding/breastfeeding-a-confidence-game/
https://www.bellybelly.com.au/Breastfeeding/breastfeeding-a-confidence-game/
https://www.bellybelly.com.au/breastfeeding/breastfeeding-a-confidence-game/
I would start by just getting your site back in order. It looks like this was not a great launch and there is a ton of cleanup that needs to be done.
If you're moving from a dev environment to live, it's not redirects that you need. You need to update the references in the code of the template and in the content areas that your developers didn't clean up when moving the site to live. Doing rewrites will only cover the issue, but the incorrect links will still be there.
While there are definitely some issues, the main problem here is simply time and links. Moz Local was a great idea but I'd also suggest claiming more social media profiles. Google has a sandbox for new sites where it is reluctant to rank them. A few months from now, with more local listings and social media listings and your site will show a lot better. Be sure to focus on some of your industry specific links as well.
Doesn't have to be so tedious. If you have a list of URLs you can use the bulk removal extension for Chrome found here: https://github.com/noitcudni/google-webmaster-tools-bulk-url-removal
Based on your clarification above about the redirects, you are okay leaving it as is, as long as everything is redirecting properly. It would still be best practice to change the redirects from page 1 > page 3 and page 2 > page 3, so all old redirects point to their newer counterparts.
Your backlink profile consists mainly of generic directories and generic article sites and almost every link is keyword rich. You most likely are hit with an over-optimization penalty from Penguin. You really need to be more careful on the type of links being built, quality > quantity.
It would be best practice to update the old redirects to the new structure as well.
It looks like somewhere around September or October 2012 something happened. There were both Panda and Penguin updates around that time. You may want to try an overlay tool such as Panguin http://www.barracuda-digital.co.uk/panguin-tool/ to see when the dips happened. You should also see if you have old ranking data for that time period or at least compare top pages to see where the traffic was lost exactly. I'm also pretty sure you're looking at total website traffic, and not just organic traffic. This could be something as simple as you stopped running a PPC campaign at that time and what you see now is the traffic without paid advertising. Also see if anything happened with the site around that time like a redesign or a change to the url structure.