[DuncanWaring]

The elevator is moving “sideways” at the same speed the Earth’s surface is moving “sideways”,

Not true.

The angular velocity is the same; 360 degrees per 24 hours, but in 24 hours an object on the surface of the earth moves 25,000 miles (the earth's circumference), but an object in geostationary orbit moves over 150,000 miles.

[B-Chan]

You're wrong. The time it takes a satellite in orbit to travel once around the Earth is called its orbital period (P). P can be calculated using the following formula:

P=2p*sqrt(a³/m)

where P = the orbital period, a = the semi-major axis of the orbit (same as the radius of the circle for a circular orbit), and m) = the gravitational parameter (~398601 km³ / Sec² for Earth).

Where P=24 hours, a=35,786 kilometers.

NASA has a handy online calculator that allows one to calculate the period of any circular orbit with an altitude >185 km. Try it!