First thing to note is that since F => S, we have P(W|F) = P(W|F,S), so we can just work out P(W|F)
With respect for your diligent effort and argument, nonetheless: Fail.
F => S -!-> P(X|F) = P(X|F,S)
In effect what Eliezer and many commenters are doing is substituting P(F|W,S) for P(F|W). These probabilities are not the same and so this substitution is illegitimate.
(Had your argument above been correct, the probabilities would have been the same.)
Conditioning on survival, or more precisely, the (continued?) existence of "observers", is just what anthropic reasoning is all about. Hence the controversy about anthropic reasoning.
To understand the final question in the post, suppose that you hooked yourself up to a machine that would instantly and painlessly kill you if a quantum coin came up tails. After one hundred heads, wouldn't you start to believe in the Quantum Theory of Immortality? But if so, wouldn't you be tempted to use it to win the lottery? ...that's where the question comes from, anyway - never mind the question of what exactly is believed.
See also: Outcome Pump
Recently the Large Hadron Collider was damaged by a mechanical failure. This requires the collider to be warmed up, repaired, and then cooled down again, so we're looking at a two-month delay.
Inevitably, many commenters said, "Anthropic principle! If the LHC had worked, it would have produced a black hole or strangelet or vacuum failure, and we wouldn't be here!"
This remark may be somewhat premature, since I don't think we're yet at the point in time when the LHC would have started producing collisions if not for this malfunction. However, a few weeks(?) from now, the "Anthropic!" hypothesis will start to make sense, assuming it can make sense at all. (Does this mean we can foresee executing a future probability update, but can't go ahead and update now?)
As you know, I don't spend much time worrying about the Large Hadron Collider when I've got much larger existential-risk-fish to fry. However, there's an exercise in probability theory (which I first picked up from E.T. Jaynes) along the lines of, "How many times does a coin have to come up heads before you believe the coin is fixed?" This tells you how low your prior probability is for the hypothesis. If a coin comes up heads only twice, that's definitely not a good reason to believe it's fixed, unless you already suspected from the beginning. But if it comes up heads 100 times, it's taking you too long to notice.
So - taking into account the previous cancellation of the Superconducting Supercollider (SSC) - how many times does the LHC have to fail before you'll start considering an anthropic explanation? 10? 20? 50?
After observing empirically that the LHC had failed 100 times in a row, would you endorse a policy of keeping the LHC powered up, but trying to fire it again only in the event of, say, nuclear terrorism or a global economic crash?