+1 to what Ross said. Subdomain hosted elsewhere won't have any impact. However, you should consider moving your subdomain to the sub-folder if that makes sense for your SEO Strategy.

Have you also seen any impact in traffic during the same timeframe?

Take a look at this post on Moz - https://moz.com/blog/seo-changes-using-google-tag-manager. Also, take a look at https://productforums.google.com/forum/#!topic/tag-manager/Ilxks3tQKrg. Personally, I wouldn't recommend doing this via GTM. Noindex and Schema specially. They are both important for SEO and are better off done via the CMS.

If your AMP is setup correctly, I wouldn't worry about this. Do you have the correct amphtml tag and the reference back to your canonical?

Eric, Have a few questions for you:

1. Do you have executive buy-in to do SEO Work?
2. How much SEO potential is out there for your website to rank?
3. How well does it rank today?
4. How big are your Top 3 competitors? Have they been growing last 1-5 years?

If you can demonstrate the size of the opportunity you have and prove it's achievable with a smaller project, test or experiment, you might have "unlimited resources" ;). Just saying :).

I highly recommend Screaming Frog on top of Moz. It can do a lot of Ad-Hoc crawls and help you diagnose potential issues. Try and play with the free version. Depending upon the site of the site, you might be okay with the less than 500 pages. However, at $200 a license, it's not much for a larger site. Imagine what a bug could cost you? 

Directory links are the "manipulated" links of yesterday. I would strongly suggest to ignore them. Great details Alick300. I would suggest to spend some time reading Moz's Link Building and SEO Guides i.e. http://moz.com/beginners-guide-to-link-building

It's very easy to get in trouble with the wrong links.

Was this a Wordpress blog and did you install a plugin to do these changes ?

Yes, it will. Also looks like custom code, it depends on how the header is coded. But it should work. Test it, if you can. This should solve your problems relatively easily. If nothing works, you can always do a robots.txt deny for /property-search-page/?* pages, but that's not a recommended solution. Try the canonical way to see if it works first.

I checked the homepage and the housing page and both have the blank description tag. It definitely looks like a plugin/theme issue. Looks like your All in One SEO plugin descriptions are showing alright. Check in your theme files, specially header.php and see if there's an empty description tag in there. You never know, it happens sometimes

I agree with Federico one hundred percent. Figure out what your primary SEO friendly URLs are for these kinds of pages and canonical them back to that page.

You have gotten some great answers. What I would like to add is, reverse the process of thought a bit by thinking where your customer is. Go list with directories where your customer might be looking for a company like yours. If you are looking to register with any directory, do it for the value of the traffic or exposure that directory might be able to pass on to you and not the SEO/link value. If we do that, you'll get the best ROI and it'll be a safer approach in terms of SEO in 2013. I hope this helps.

I totally agree with the suggestions from Marissa and Tim. You obviously understood the problem and were concerned, therefore you posted the question.

Definitely only build 1 page for these 3 keywords. I love Tim's title suggestion.

Managed IT Services and Support | Company Name or

Managed IT Services and Support by Company Name

You are good to go.