To your point 1, arguments of this sort are not that we can't make a computer, someday, that will encapsulate consciousness, it's an argument that we will never be able to. People will try to use fancy contortions to get around it, but the basic argument is that consciousness is fundamentally not "of this reality", so no, the article is squarely taking a stance that thinking machines are not a "maybe, someday" event, they're a "never" event.

To your point 2, this directly contradicts your point 1, and is, what I would call, a "dualist" philosophy. This is a basic statement that thinking, the way we do it, is fundamentally not mechanical and therefore not describable by our laws of physics. That is, however we think is literally not of this universe. The "dualism" comes in by naming the other "plane of existence" (or whatever you want to call it) as the companion universe or reality to ours where our intelligence effectively resides.

I think Turing is a little maligned here. His test was meant to strip away all the red herrings that one might use and focus on what (I'm imagining) he thought was the most fundamental of all human traits, the ability to reason and to use language. The Turing test is a functional one. If something talks like a human, reasons like a human, acts like a human, then how would you tell the difference between the "real" human and the "simulated" human? By creating a functional test, it gets at the core of the issue. If you can't come up with a test to differentiate one from the other, the question of what's "really" human becomes academic.

So, to, the focus on Turing machines. The specifics are a little byzantine, nowadays, but the point is that there's an "equivalence" of machines, where one can be used to simulate another. The point about being Turing machine equivalent is the mechanization part, not the specifics of tape reading and what the state machine looks like, so much.

In terms of proof, I would say this: If someone were to predict the progress of machine learning and artificial intelligence as it progresses towards becoming feature parity with the human brain, one might well predict all of the milestones that have been achieved. For example, beating the best human at checkers/chess/go, being able to do speech to text, being able to identify people by faces, being able to replicate someones voice/gait/face, being able draw novel art, being able to hold on a conversation, etc.

In my opinion, the human brain is not monolithic and is compromised of many sub systems interacting with each other. That being the case, I would expect some type of "general intelligence" to come by once all the sub systems have become modular and easy to use, with some kind of "operating system" bringing it together to be able to have one part communicate with another and be able to make broader scoped decisions based on input from each of the subsystems.

This is a long way of saying I believe there is quite a bit of evidence that we're getting closer to "thinking machines". They may not work exactly how our brains work but that's kind of beside the point, they're functionally doing what we would expect them to do on the path towards human brain feature parity.