what is the average linear speed in miles per hour for a spot on the equator of the earth?
Depends on your point of reference.
The equatorial diameter of the earth is 7926 miles so in a day this spot travels the earth's diameter or 24,902 miles. Dividing by 24 produces 1,037.6 MPH. This is the speed of your spot relative to the Earth.
The earth orbits the Sun at a distance of 93 million miles so in a year it travels 292 miles dividing by 365.25 and 24 gives us 33,330 MPH relative to the Sun. Of course add or subtract about 1,000 MPH depending on if it is night or day.
Our entire solar system travels within our galaxy the Milky Way. Probably the entire galaxy revolves around a black hole that we are 26,000 light years away from. Our solar system orbits of the galaxy is about 82,000 light years and takes somewhere between 220 and 225 million years to complete. Relative to the center of the Milky Way the Earth travels 558,000 miles/hour.
Astronomers know that our Milky Way travels within our local group of galaxies about 666,000 miles/hour.
Our entire cosmos may be in motion relative to other unknown cosmos but of course we would have no way to know it.
Use the equation v = wr
v = 2pi/24 * 3 963.168
v = 1040 m/h (to 3SF)