As Leonie states, the search engines are for public facing content. If your site is completely private then you'd be more interested in making sure it's not found anywhere other than by members, however it sounds like you have some aspects of the site that could be public or created to attract new members. Typically in these cases you pull small topical samples from the site that are shown to benefit the members and help articulate why membership is valuable.  It may be a matter of having what is practically like two sites: the public facing, membership recruitment site, and the private, non-indexed membership site.  Cheers!

Google itself bought the exclusive rights to the .app TLD so there's that in the custom TLD column, but like you and others say here, .tv isn't one like that.

Finding the exact numbers on this will probably be unlikely without access to their analytics, but you can get a guesstimate by seeing which of their pages are used as site links when you search for their brand in Google. You can also derive some guesses via search rankings by dividing out a percentage of clicks based on ranking from total search volumes.

Still, you're going to be missing a lot of data by not knowing how well they're doing in terms of email marketing, social, referrers, etc.  Companies that try to accomplish this are ones like Compete and Quantcast.  A better way to go about finding popularity by topic though would be to use a tool like Google Trends, Twitter Trending, etc.

As for time on page, why?  You'll be better off focusing all your efforts there on your own site's usability, experience and customer satisfaction.

Within Webmaster Tools you could possibly exclude the subdomain pages from being listed as site links, but Google has stated they're trying to get better at associating when subdomains are similar to subfolders on a site. Rand recently discussed that aspect of it here: http://moz.com/blog/subdomains-vs-subfolders-rel-canonical-vs-301-how-to-structure-links-optimally-for-seo-whiteboard-friday.  Cheers!

None that I know of. Most of the optimization tools available can be used on any domain. See: https://support.google.com/webmasters/answer/80472.

Video content includes web pages which embed video, URLs to players for video, or the URLs of raw video content hosted on your site. If Google cannot discover video content at the URLs you provide, those records will be ignored by Googlebot.

A .tv won't enhance the ability for Google to find your videos, but proper tagging and sitemapping will. Cheers!

Right. That's due to updating after the Open Site Explorer update. Once Moz recrawls your site it will apply the changes and you'll see the numbers as they were previously.  It sounds like you've made these changes VERY recently. Any redirection of pages (even if they're on the same domain) take a bit of time to be fully recognized by search engines and Moz's OSE.

How recently have you made these changes? Have you also utilized the Change of Address tool in Google Webmaster Tools?  PA changes per OSE update, so if you ran these changes after the last update, you won't see new numbers until the next release: http://moz.com/products/api/updates (March 11th)  Cheers!