If you are inside the sphere of influence of Earth (See: https://en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics)) , You will be pulled towards the center of gravity. If you don't want to come close to earth, you need to balance the pulling force in the opposite direction. As you've mentioned above, You need to create centrifugal force. Today, it's created by following (generally circular) orbital path around the center of the Earth. The orbital speeds depends on how close you are to Earth. Closer you are, faster you need to go on the orbit. Vice versa. Normally if you stop moving in the orbit, you start moving towards Earth. This happens whether
you are over equator or n/s pole.
The reason they propose for the elevator to start somewhere on equator, has to do with geostationary orbits. Satellites on geostationary orbits, need slower speeds to achieve orbital balance since they are farther than the satellites at low-earth-orbit/LEO. In fact they have speeds matching Earth's rotational velocity. This gives the illusion that they are hovering always in the same spot but actually they are moving at their orbit slow enough to match Earth's rotation.
Since Space Elevator's structure will expand from surface/some-point-in-atmosphere all way to space, you need to put some balancing mass (space-platform?) at the space-end to create sufficient centrifugal force. If you put balancing mass/space-platform at low-earth-orbit / LEO, It has to go fast enough on orbit to balance gravitational pull. It would be similar to other satellites at LEO meaning It would go around the earth around 90 minutes or so (just like ISS). Imagine a giant , long elevator shaft moving 7.8 - 8 m/s in the atmosphere . Never mind the air-resistance, It would be impractical to have an elevator shaft moving fast like that.
Instead they came up with the idea of placing the balancing mass/space-platform at around geostationary orbit in order to reduce relative motion against the Earth to minimum. The relative motion is at the least when elevator shaft starts on equator and go all the way up to balancing mass at geostationary orbit. This way "Balancing mass-elevator shaft" system moves very very slowly relative to Earth and you would have a stable Space elevator. Of course current technology it is not sufficient to create physical structure which can withstand even its own weight due tensile stresses generated in such a massive structure.
An alternate method proposes of putting the balancing mass at Medium Earth Orbit / MEO and putting starting point of the elevator at the upper atmosphere to reduce air drag and avoiding potential hazard due to the moving elevator platform. But then there is the ever-present hazard of colliding with satellites at LEO and/or MEO since this system would go a lot faster then the equator-positioned model. There was a Space Elevator contest in 2010 with 2 main competition fields: Cable Tensile Strength, Powering the elevator/carrier device. you should check it out : https://en.wikipedia.org/wiki/Elevator:2010
For your last question about using centrifugal force due to earth going around Sun, I'm not sure on this one, I'm just gonna give you my guess. As long as you are inside of earth's sphere of influence, you will be affected by Earth's gravity for the most part; so centrifugal force created on earth due to earth's orbit around the sun would not be helpful ,in the slightest, to build a space elevator for earth. That force only keeps Earth from falling to the Sun. You need to overcome primarily the Earth's gravity.