What are the motions of the celestial sphere as seen from the Earth's equator and poles?

Answer 1

Right now, Polaris is 1 degree away from the North Celestial Pole. There is no Southern Star right now. The Earth wobble on its axis every 10,000 years or so. So in a few thousands years from now, there will be a new North Star and maybe a South Star. From the North Pole and South Pole, all constellations are circumpolar, meaning they never rise or set. The farther North or South you go, there will be different circumpolar constellations and many more constellations will rise and set. At the equator, all constellations rise and set.
All non-circumpolar constellations rise in the east and set in the west.

Answer 2

The celestial sphere seems to rotate around the celestial poles, which are directly above the Earth's poles. Of course it is an illusion. The celestial sphere is stationary and Earth rotates.

Answer 3

The poles are always pointing in the same directions at any given time during the year...if you stood directly on top of the north pole, the North Star (Polaris) would ALWAYS be directly over head...with everything else in the celestial sphere rotating clockwise around it--east to west. The Sun, Moon, and planets are along a line through the sky that encircles our planet that we call the ecliptic...this is the plane of our solar system....so anything part of our solar system would appear to move along a line from the vantage point of Earth against the further background stars. The only exceptions to this are comets and the moons of other planets...the moons tend to lie in a plane that's congruent with the given planet's equator...such as Saturn, whose moons are in the same plane as its rings, which shift relative to the ecliptic. (The one exception?...OUR OWN MOON!)
From Earth's equator, the motion depends on what time of year it is...since its axis is shifted on an angle against the ecliptic, you'll see different constellations at various times of the year.