However, it can pass some rank and power indirectly.
PageRank can only be passed directly from one site to another. At it's core, a lot of PageRank is related to a citation. A citation is when a resource is publicly referenced. For example, if ten people are in a room and person X is asked by person Y who cooked the Scungilli, the only way to identify that person is to point/nod/look at that person. You can't indirectly gesture to a person. The same is true with citations.
There are other things in play related to user intention
that can seem to override links, anchor text, on page factors. But that's an entirely different discussion but it wouldn't surprise me if that's what is throwing off your sources of information and causing them to chase down red herrings.
...and the fact that you rank so well in another search engine can pick up with Google.
Afaik, and correct me if I'm wrong, but it has only been the case that Microsoft has been caught poaching/copying Google's SERPs and not the other way around.
...and big g uses this information constantly.
The Google Analytics data is supposed to be used only in aggregate. Google can obtain specific site data from other sources apart from Analytics.
According to several sources, no.
Those kinds of "studies" make great link bait but don't ever help anyone rank and are frequently debunked. There is another layer of algorithm being imposed on the SERPs related to user intent, which can cause your sources of information to chase red herrings. People who make those kinds of unscientific link bait statements are regularly proven wrong, their methodologies are often shown to be unscientific or miscalculated through a shaky grasp of math or statistics.
Those kinds of people generally have a history of being wrong about a great many things, from what link selling businesses they publicly endorsed (and now do not endorse), to self-serving statements about Google's ability to spot certain kinds of link manipulations. Study their background, what their positions used to be about certain issues related to link building and how they changed their tune only when it became glaringly obvious. Most of those kinds of people have a history of being wrong, a history of doing what was expedient at the moment. No need to expect them to be correct now. Their history of getting it wrong is why those kinds of people are often in the business of selling tools. It's profitable to sell shovels to would-be gold diggers.