Do pilots need to know how to use a sextant?
Ooh, a celestial navigation question...about airplanes... I never thought I'd see one of these in the 21st century!

To answer the basic question: Pilot training doesn't cover sextant use, but there is still a "Flight Navigator" certificate (you'll find it in Part 63 of the FARs), and an associated Flight Navigator Handbook (FAA-H-8083-18), last updated in 2011. The written test question bank is also available if you want an idea of what the FAA expects you to know in order to earn that certificate.




  Can a sextant be used while flying?


Yes, if you have a clear view of the sky overhead.
Aircraft used to be designed with "sextant ports" as shown in Stelios' answer, and special aircraft sextants (indirect-sighting, with a bubble to provide a horizon, and usually some kind of averaging mechanism and timer).




  How accurate/reliable is a sextant, both standing still and at 500+ mph? 


A sextant is a pretty accurate tool under the right conditions, if you know how to use it. 
Stationary on the ground you can do a surprisingly good job with a sextant, see for example the US/Canada border monuments. I'm out of practice, but on calm water I used to be able to fix my position to within about 1 minute of arc (1 nautical mile) on any given line.
To put that in perspective, that's slightly better navigational performance than a VOR provides at distance (2-4 nautical miles), but substantially worse performance than GPS (within a few feet).

As far as reliability, the reliability of a ship's sextant is pretty good: much like a mechanical E6B flight computer it'll work fine unless you abuse it (drop, step on, etc.).
Reliability of an aircraft sextant (or any other bubble sextant with an averaging timer) is not quite as good because there are more moving parts involved, but properly maintained they could be trusted to work when you needed them (and most failures could be compensated for by a decent navigator).  

The problems with sextants that make them impractical for modern air travel are:


Speed
Sextants don't tell you where you are, they tell you where you were when you made the observation. The averaging mechanisms I mentioned earlier allow aircraft sextant to partially account for the fact that the aircraft was moving (both vertically and horizontally), but at modern speeds you cover a minute of arc pretty quickly. Radio navigation systems and GPS provide much faster position updates than you could get with a sextant.
Complexity of the instrument
Aircraft sextants are precision instruments. They need to be maintained and calibrated or they won't be accurate when you need them, and as I mentioned there are a lot more moving parts to an aircraft bubble sextant than a basic ship's sextant.  
Complexity of the process
Using a sextant also require an operator (navigator) who knows what they're doing in order to take the sighting and compute the position, compared to radio navigation (VORs, VOR/DME, or GPS).
Most pilots don't have any training in using a sextant (my experience comes from sailing), so having one would be useless to them.
Weather
While the sextant is a reliable instrument the stars are not reliable: They're always there, but sometimes they're not visible. You can't see most of the stars &amp; constellations you would take a sighting of during the day (usually only one star is visible), and clouds can obscure them when you want to take a sighting, delaying or preventing your position fix.





  Are sextants found in any cockpits today?


Certainly not in any light GA cockpit I can think of, and probably not in any single-crewmember cockpit: Even if you held both a pilot certificate and a flight navigator certificate a pilot with their head buried in the sextant is rather useless for actually flying the plane.

I would assume that most commercial cockpits also no longer have a sextant as modern jets don't include a sextant port, so there's no way to take a sighting.

There might be a few restored vintage aircraft that have a working sextant onboard, but I doubt those are used for practical navigation (they're probably not used at all, except possibly to demonstrate how it was done).