What planets can be seen in certain constellations?
The constellations of the zodiac fall on the plane of the ecliptic (the plane in which the Earth orbits the Sun) and all the Sun's planets orbit also in the plane of the ecliptic. This means that ALL the planets eventually move through ALL the zodiacal constellations (as seen from Earth). It also therefore follows that the planets are never seen in the non zodiacal constellations.
Why do the planets appear to move through the constellations instead of with the constellations as Earth rotates?
There is an imaginary band in the sky called the Plane of the Ecliptic. The planets we can see from earth, never wander above or below that band of sky - so the planets are always in one of the other of the constellations that lie in that band. We call those particular 12 (or 13) constellations the "constellations of the Zodiac".