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Some years ago people used to make this claim about W3C validation badges too. A badge is a badge is a badge. It really is unlikely to affect your search rankings one way or another.

Due to the extraneous text at the bottom of your question I am answering each question with the subject of the question prepended to the answer in parentheses. So in answer to your questions:

1. (tools available) Moz Analytics, Followerwonk, Open Site Explorer, Fresh Web Explorer, Rank Tracker, Keyword Analysis (including keyword difficulty), On-Page Grader.

2. (crawl entire site) Yes, add your site as a campaign and Moz will crawl your site and inform you of any problems and factors affecting your rank. It will also track changes over time without manual intervention.

3. (10 campaigns) Yes, each domain is a separate campaign.

4. (subdomains) This is configurable. You can set in campaign settings whether to track only this subdomain or all subdomains or even just a specific sub-folder.

5. (keywords) The number of keywords relates to how many keywords can be tracked in your account as a whole across all of your campaigns. 750 over 10 campaigns gives you 75 keywords per campaign. The rank of your site's pages is tracked for each keyword, pages are graded for their ranking ability against specific keywords and you can check the rank of any page for any keyword you are tracking.

6. (social accounts) You can track the performance of your social media accounts by connecting them with Moz Analytics. You can also track your competitors accounts to measure your performance against competitors in some aspects such as the level of interaction (comments and shares).

7. (branded reports) Branded reports allow you to export reports on your data with your own branding, a useful feature for agencies and consultants.

I am a Moz Pro subscriber and highly recommend it, Moz Analytics and the various other tools are extremely useful.

Your site is competing in a very highly competitive field. I did a check in Moz Keyword Difficulty on your example page's targeted keyword (nokia lumia 830 review) and that keyword is highly competitive. The Google India SERP for the keyword is dominated by high domain authority sites with high page authority pages.

I'm afraid it is the obvious. Ensure the rich snippets are relevant to the content of the page, ensure that your page is ranking for relevant queries to raise the chance of the rich snippet being shown.

Google only shows rich snippets when it thinks it will be useful to the searcher.

As you have said you have had some issues with maintenance, check the structured data against Google's structured data testing tool. However it is more likely that Google isn't showing the rich snippets because it believes your page quality to be low or that the structured data is not relevant to the user's query.

Great to know that Moz is working on support for other countries!

Moz Local presently only supports the United States. I ran into this myself a while ago. Hopefully they'll get around to supporting UK and Germany soon.

Very interesting! I don't recall seeing that before but I checked the Internet Archive's Wayback Machine entry for that URL and the quoted extract has been there since at least 2013.

Elsewhere Google has been pretty insistent on structured data being part of the document itself as much as possible so it does seem somewhat contradictory advice. As you say perhaps they've simply forgotten to update that particular entry to reflect current thinking.

Once upon a time it was possibly a good use of the meta description to include some salient structured data but today we have a proper way of marking up structured data. The meta description is best used for compelling, relevant copy to attract the user to click through to your site as the meta description is your one best hope of affecting what is shown to the user in the SERPs.

Search engines haven't shown any inclination to parse the meta description and I doubt they would do so in future. Structured data belongs in the document itself, marked up accordingly.

In 2011 Matt Cutts advised that Google does limit on redirect chains - he indicated the Googlebot won't follow more than around 3 or 4. No limit on the total number of single-level 301s.

In your specific situation I would redirect both the original and the first replacement to the new replacement so that users and bots can reach the new page in a single redirect hop.

Your uploads directory is viewable. I would configure the hosting to disallow browsing directories.

Because WordPress creates a directory for each month you have two dozen pages being indexed by Google that show nothing but filenames ending with the .png extension. This is most likely where the keyword in Webmaster tools is coming from.

Due to the extraneous text at the bottom of your question I am answering each question with the subject of the question prepended to the answer in parentheses. So in answer to your questions:

1. (tools available) Moz Analytics, Followerwonk, Open Site Explorer, Fresh Web Explorer, Rank Tracker, Keyword Analysis (including keyword difficulty), On-Page Grader.

2. (crawl entire site) Yes, add your site as a campaign and Moz will crawl your site and inform you of any problems and factors affecting your rank. It will also track changes over time without manual intervention.

3. (10 campaigns) Yes, each domain is a separate campaign.

4. (subdomains) This is configurable. You can set in campaign settings whether to track only this subdomain or all subdomains or even just a specific sub-folder.

5. (keywords) The number of keywords relates to how many keywords can be tracked in your account as a whole across all of your campaigns. 750 over 10 campaigns gives you 75 keywords per campaign. The rank of your site's pages is tracked for each keyword, pages are graded for their ranking ability against specific keywords and you can check the rank of any page for any keyword you are tracking.

6. (social accounts) You can track the performance of your social media accounts by connecting them with Moz Analytics. You can also track your competitors accounts to measure your performance against competitors in some aspects such as the level of interaction (comments and shares).

7. (branded reports) Branded reports allow you to export reports on your data with your own branding, a useful feature for agencies and consultants.

I am a Moz Pro subscriber and highly recommend it, Moz Analytics and the various other tools are extremely useful.

In 2011 Matt Cutts advised that Google does limit on redirect chains - he indicated the Googlebot won't follow more than around 3 or 4. No limit on the total number of single-level 301s.

In your specific situation I would redirect both the original and the first replacement to the new replacement so that users and bots can reach the new page in a single redirect hop.

Once upon a time it was possibly a good use of the meta description to include some salient structured data but today we have a proper way of marking up structured data. The meta description is best used for compelling, relevant copy to attract the user to click through to your site as the meta description is your one best hope of affecting what is shown to the user in the SERPs.

Search engines haven't shown any inclination to parse the meta description and I doubt they would do so in future. Structured data belongs in the document itself, marked up accordingly.

It is absolutely fine to reach out and gently ask for a correction, in my opinion. I have done it myself several times recently and all individuals contacted were quite happy to oblige me. The important thing is to be sincere and genuine in your message and remember that the person on the other end is likely busy so keep your message to the point and give precise instructions to minimise any need for back-and-forth email clarifications. Be patient - but chase after a month or so, if necessary - and do follow up to thank them for correcting once they have done so.

Some years ago people used to make this claim about W3C validation badges too. A badge is a badge is a badge. It really is unlikely to affect your search rankings one way or another.

Google only shows rich snippets when it thinks it will be useful to the searcher.

As you have said you have had some issues with maintenance, check the structured data against Google's structured data testing tool. However it is more likely that Google isn't showing the rich snippets because it believes your page quality to be low or that the structured data is not relevant to the user's query.

The cache info box shown in Google's cached pages is actually present but is being hidden by the website's design, specifically the div element with the classes "wsb-canvas body".

I think it is important to distinguish the purpose of the automated validation services offered by Google and Yandex, which is to ensure that the properties utilized by the respective search engine are present, and the actual schema.org structured data initiative which doesn't place many requirements on publishers.

With that in mind when Yandex states that the address and telephone properties are required for http://schema.org/Organization, it doesn't mean they are required by schema.org but rather that they are required by Yandex. Google's Structured Data Testing Tool doesn't state that these are required because for Google's purposes they are not.

So both are correct but for purposes of ensuring your structured data is showing up you do need to test in all of the relevant tools. For less mission-critical structured data it is OK just to follow the Schema.org documentation and wait for the providers to implement support.

Implementing SSL should be straightforward for the most part

You need to ensure that links around your site (including canonical links) are updated to use HTTPS (so https://example.com/link as opposed to http://example.com/link where example.com is your domain name). If you are already using a protocol-less linking pattern (//example.com/link) you don't need to update the links.

You can also configure your web server to only serve HTTPS. If your web server is Apache you can do this with the SSLRequireSSL directive.

 <code><location>SSLRequireSSL</location></code> 

HTTPS also causes a significant slow-down as the browser and the server negotiate a secure connection. If your site has already been optimized for speed it should not cause a problem but if in doubt revisit that process and ensure that you are getting the best possible speed for your visitors.

The article by Cyrus has a great checklist to double check everything.