Just to return to the part of the discussion in which Dawkins claims that there is no free will. A very great deal depends on his having assumed mind-body (Cartesian) dualism. But it requires philosophical work, which he is either not prepared or not capable of doing, to understand its influence.
Anthony Kenny: Most philosophers don't like the naive picture of free will that that experiment presupposes ... that there is a soul inside in which mental events occur that are the causes of bodily events. ... It's surprising that you [Dawkins] should accept it [i.e. accept the experiment and what it 'shows'] because it's very much the 'ghost in the machine' picture [Dawkins tacitly accepts the mind-body duality picture but denies there is a ghost]. You say, "Ah! The 'machine' works before the 'ghost' does", but I think most philosophers today believe that the whole idea of constructing mind and body like that is quite wrong."
RD: "But why doesn't it destroy the idea of free will?"
AK: "Because it only shows the order of events in an act that is undetermined may not be what you would have expected if you had the false philosophical idea."
The experiment presupposes, and 'tests for', the mind-body duality - the false philosophical idea. The experimenter anticipates mental activity giving rise to physical activity, and looks closely at the brain (where it is supposed the immaterial mind must be seated), then finding that the physical activity precedes the brain activity concludes there is no mind.
Only someone who holds the mind-body duality to be true would attempt to find evidence for mind with this kind of experiment - as if 'chopping up' the brain will somehow show "Aha! You see, there's nothing extra there!" It is very odd that one should look to demonstrate the non-existence of something immaterial by examining material as minutely as possible in order not to find it.
I find brief and sensitive - and humane - reflections on the incoherence of Cartesian dualism (and the un-knowing perpetuation of it in para-science writing) in Marilynne Robinson's "Absence if Mind - the dispelling if inwardness from the modern myth of the self". See the chapter "Thinking Again".
See essay by Dr Rowan Williams, former Archbishop of Canterbury, here: http://rowanwilliams.archbishopofcanterbury.org/articles.php/1211/analysing-atheism-unbelief-and-the-world-of-faiths