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I've see statistics showing that scientists tend to be less theistic than the general population and that the best scientists (National Academy members, for example) tend to be less theistic than scientists in general. So that provides the correlation you are asking for. But, I strongly suspect that in this case, correlation does not imply causation.
I have seen numerous examples, though, in which scientific enquiry with the choice of subject matter motivated by theism is of lower quality than science done without that motivation. However, the same kinds of bad results can arise from motivation by social activism or personal animosity or simply prideful intransigence.
Absolutely. Hence, the warning flag. A scientist expecting to find the evidence of God doesn't just have freeloading beliefs, but beliefs that pay rent in wrong expectations. That's akin to a gambling economist.
I'd say it's good evidence in favor of P ( good science | scientist is theist ) < P ( good science ) . Of course, your point about correlation not causation is very valid, too.
Someone in the discussion once said that atheism on average adds ~40 to IQ (I might be remembering incorrectly). I suppose high IQ is correlated with both excellence as a scientist and an ability to reconsider and abandon theism if the question ever arose.
My specific interest is whether or not atheism alone makes scientists better.