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The key point is that it's exceedingly unlikely for all of Turkey to get slapped with an "avoid all travel" advisory, but I'll answer anyway for posterity. One by one:
Airlines can choose where they fly irrespective of travel advisories, and there are commercial flights to places like Baghdad and Mogadishu that no sane voluntary traveller goes to. So, no, the State Dept saying "don't go" doesn't mean that your flight in or out will be cancelled. If anything, airlines know money when they see it and they tend to add flights to places people want to get away from at any cost.
Your exact rights in the event of involuntary flight cancellation are complex, depending on the contract of carriage, the country you're leaving from, the airline etc. Most flights from the US to Turkey would go via the EU, in which case the EU's policy (which is, fortunately, unusually clear and well-enforced) would apply. In short, you would be entitled to your choice of a full refund or alternative transportation to your destination (eg. another airline that's still flying). However, since war is an "extraordinary circumstance[s] which could not have been avoided even if all reasonable measures had been taken", you would probably not be entitled to the monetary compensation you would usually get for an ordinary cancellation.
Many better travel insurance plans cover involuntary flight cancellation. However, virtually all travel insurance plans have weasel wording that gets them out of anything related to war or civil strife that the traveller could have known about (= travel advisories), so read the small print. That said, since you're already reasonably well protected on most airlines, I wouldn't go out of my way to find insurance on top.
If you change your own plans, even though the airline is still flying, then that's your call and you'll need to negotiate with the airline to rebook your flights. I'm not aware of any travel insurance policy that would cover this for you unless it's due to some factor outside your control (death or sickness in close family, loss of job, etc).
If the excrement really hits the rotary ventilator, yes, the State Department will try to evacuate its citizens. (That's assuming they know you're in the country, which they won't unless you tell them, so sign up here if you want Uncle Sam to watch out for you.) However, this is very far from a magic carpet ride out, they'll charter a plane at full commercial rates -- which are pretty darn high for one-way trips out of war zones -- and send you the bill, which can easily be in the thousands: after the Haiti earthquake, evacuees were slapped with bills for $1,600 for the 45-min hop to the Dominican Republic, and that's on the cheap side. So unless you're in Saigon in April 1973 and the Viet Cong tanks are rumbling in, it's generally a much better idea to find your own way out if at all possible, and from Turkey this really shouldn't be a huge problem (flight on another airline, train or bus to Greece, ferry to northern Cyprus, whatever).