What really sums up the more significant aspects of the question of intuition is this part-
"Intuition could be described as the opposite of self-consciousness while at the same time it is probably the most conscious way of experiencing a lively moment..."
I sometimes think It's a way of tapping into greater consciousness not individual consciousness.

Talking about the guessing the cards, it's so difficult. I've done similar experiments and the moment one becomes invested in desiring a certain outcome , one looses the ability to do so. I think this is what sucks gamblers in, and keeps people playing the lottery. They can feel something is there, some power but the very 'need' to win always obscures the power. I did some research into the circumstances surrounding lottery winners, trying to find some pattern or a common thread and the one thing I did notice was the situation surrounding many winners: Whether the y were regular players or random or one time triers, there was often a forgetfulness'. They forgot they bought a ticket or forgot to check or forgot their usual numbers and went for a quick pick until a nagging sensation kind of hinted at them to check, so they found they were winners. I tried to do this, but it's hard to remember to forget ;-)

I can relate with the story of ball playing. The time I played my finest golf was when me and a friend just walked casually around a course carrying only one or two clubs. Usually I would play with my father and brother where there would be some pressure and a lot of advice. Just casually swing the club and not expecting to much (in terms of distance and accuracy) resulted in much straighter even shots. Maybe not as far but never off into the bushes...

I think part of the reason sports professionals (especially in things like billiards or darts) are able to cross that tipping point between being an able amateur and a 'talented' player is because they enjoy it and so end up playing it so often that they can afford to be 'casual' about it 70% of the time and thus develop that intuitive sense in a relaxed atmosphere. People who only get to play 'occasionally' will pressure themselves. Pros have time to just 'jam'. How they generally do not make use of the full potential is to keep this sense limited to their particular field without realizing it can be universally applied to improve quality of life and performance in all spheres including 'emotional intelligence'

The 'stared at from behind' phenomena has fascinated me since a child. It's so real and I've 'proven' it so many times. It must be universal to human experience yet is almost completely ignored by society in general. the implications are immense. On one level I understand why people don't want to think about the whys and hows concerning it. It's too mind boggling. Nice to know Sheldrake is looking into it. It's a bit like the placebo effect that science just refuses to investigate - what it means is we heal ourselves with positive thinking- yet there is no research as to how to practically harness the power. What could we do if we fully understood that we could sense attention without use of our '5 senses'. It hints at so much possibility not only about sense it but about giving it.

The differentiation with 'instinct' is an important one. People who don't like to think about these things like to just lump them together. Such a great read, you deal with the subject matter so comprehensively.
Your conclusion I think conveys an important message of remaining grounded, empirical, calm and pragmatic when investigating or working with subtle senses. To become overly imaginative think fantastically about them actually betrays an underlying lack of faith in a greater reality.