Is energy always conserved?

No!  That's all wrong!  Gravitational waves, warping spacetime in general, does not change the laws of physics.  Changing the stuff, even spacetime itself, does not change the *rules*.  

Changing the arrangement of stuff will transform energy, but is clearly conserved: if you put everything back, you return to your original energy too.  In this case, consider what happens if you shrink the universe again: you blue-shift, exactly reversing the loss with a gain.  The "change" (growing universe) encodes the potential energy lost to the photon.  

That's as if you said that throwing a ball upward makes energy disappear, not conserved, because it slows down as it rises.  It actually trades off kinetic energy with potential energy of its higher position, and is an example of exactly how conservation works!  The expanding universe is *exactly* the same case. If Sean Caroll (mentioned in the description) doesn't give you the answer, try Laurence Krauss, who has discussed this specifically.

This video is embarrassing wrong.  Please run it by a physics professor.  Even though the red-shift is kind of mysterious, you completely missed the point of Noether's Theorem as it applies to this problem.