Throughout the year, are all parts of the night sky visible from any given point on Earth? Or are there points on Earth where certain parts of the 360° night sky are never visible throughout the whole year?  

Depending on the answer to this question, are there points on Earth where parts of the celestial sphere as such are never in the sky, no matter if it's a day or night sky?
At any one time, only half the sky can be visible, and half is below the horizon.

At the poles the sky appears to rotate about the zenith, and so no new parts of the sky ever become visible.  At the North pole, you can never see any stars in the Southern hemisphere.

On the equator different parts of the sky become visible and over the year all the stars can be seen (but stars very close to the pole will be very hard to see, as they are too close to the horizon.)

For someone who is at (for example) a latitude of 40 degrees North will not see any stars that are more than degrees south of the equator; 90-40=50; and so won't see the southern cross, the Magellanic clouds or other features of the Southern sky. Similarly an observer in Tasmania won't see Polaris, Cassiopeia, or the "big dipper" as these are features of the Northern sky.