A Simple Demonstration of Space/Time Relativity

[BryanS]

And is there no connection between radial speed and time dilation?

Not in particular, no. Only if the magnitude of its linear velocity approaches a value that is large relative to the speed of light (3.0 x 10^8 m/s).

magnitude of linear velocity = angular speed (measured in radians/s) times radial distance from center of rotation
v = w * r

Two objects with the same angular speed can have vastly different magnitudes of linear speed. Think of being on a merry-go-round on a playground. If you position yourself just off of center, you are not moving very fast, but if you position yourself near the edge, you are moving faster in space. Two objects on the same spinning merry-go-round at different radial locations will have different speeds. 

The principle of special relativity would tell you that a merr-go-round that is spinning such that its outer edge is moving at speeds near the speed of light will see the outer edge time passing slower than another location near the center. But that has to do with the difference in linear speed, not in particular how fast it is rotating. It would be possible for two spinning objects with the same angular speed but different sizes to result in the larger object experiencing relativistic effects while the smaller one does not. It all depends on how big radial distance "r"  is.

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I should add that when you discuss "space time" relativity, you are referencing Einstein's special relativity, which is a lot more complicated than your apparent understanding of the concept. Special relativity relies on two assumptions:
1. The laws of Physics do not change based on space and time frame of reference.
2. The speed of light is always constant regardless of your space and time frame of reference.

You appear to be incorporating some understandings of Newtonian relativity. This is the sort of relativity that says if you are moving in a car that goes 10m/s and throw a ball ahead of the car at 5m/s, the ball appears to a person on the ground to be moving 15m/s.

Einstein proposed that for light, a person in a car moving 10m/s shining a flashlight sees light moving at 3.0x10^8 m/s. The person on the ground sees the exact same speed of this light being 3.0x10^8 m/s, which would violate Newton's understanding of relative motion. Everything else in relativistic physics, including that E=mc^2 equation many associate with Einstein, is derived with just the two above assumptions. But to see any relativistic effect at all, the number sqrt( 1 - v^2/c^2) has to be significantly less than 1. So no, a human is not capable of physically rotate a piece of paper and achieve anything close to the speeds necessary to observe special relativity. I somewhat doubt humans have created a mechanical device that could even do that.