You also can check for HTTP header results for crawling too:
https://developers.google.com/webmasters/control-crawl-index/docs/robots_meta_tag

Also you can use some of Google services for this. Specially PageSpeed API:
https://developers.google.com/speed/docs/insights/v2/reference/

Once you call this API it return JSON with list of blocked resources. It's little bit slower but i found that this is safe. Some hostings have IDS (intruder detection systems) and when some crawl them little bit aggressive they block whole IP or IP range. I know few cases when site is OK to be seen from users, but blocked from Google IP. Webmasters wasn't happy when they discover this. They call hosting few times and got "there isn't issues from our side, we didn't block anything". And 6 hours later they get "seems that another department was blocked this server for few specific IPs".

About checking for logged/nonloged users. You can use StructuredData Testing Tool. Also one call to get JSON with full HTTP response and then compare it with your result.

I just browse via Google.IT for "vangogh" and you're on N:2 spot after Italian Wikipedia article. I strongly recommend you to create account in SearchConsole and verify your site. There you can see in SearchTraffic exactly what position you get in SERP results.

Now answer for browsers - modern search engines return SERP pages optimized for specific browsers. That's why it's possible browser X to see some results, browser Y to get other results. There is no way to escape this unless you change "user-agent" string in browser with some plugin. Some engines are even going further that return personalized results based on IP, previous searches from this IP, previous searches from this account, previous clicks on SERP, etc.

The only way to get right position is with SearchConsole or special tracking tools as Moz RankTracker. And there you can see "average position".

Yes, there are 4 Schema.org objects:
https://schema.org/SubscribeAction
https://schema.org/RegisterAction
https://schema.org/JoinAction
https://schema.org/FollowAction

There is only one issue - on all of them there is "agent" that we're talking about. Like "John join ACME", "John registered to ACME", etc. And without agent codes can be verified w/o issues but look meaningless "join ACME", "registered to ACME", etc.

I also try to implement Join/Subscribe/Register as SiteLink:
https://developers.google.com/structured-data/slsb-overview
but can't pass validation because "query-input". If exist it's unknown for subscribing, if i delete it it's missing and required. So consider both ways as broken currently.

Anyway - you can markup emails:
https://moz.com/blog/markup-for-emails
and make them little bit actionable.

Links is "All links to this page including internal, external, followed, and nofollowed".
It's same number as if drop this URL into OSE where it's count as "Total Links"

This also happens in other social networks - Twitter, G+, Pinterest, LinkedIn, etc.

So if some app there opens web inside of app - this is "Safari (in-app)" in GA.