When is a circle not a circle? Imagine a 2D uniform circular motion of constant magnitude but changing direction in an area of zero g. The forces will be equal all the way round - it will be a perfect circle.
Now imagine the same circular motion on earth with the 2D circle 'upended'. If the circle described were to still be a perfect circle, considering just the effect of gravity and ignoring atmosphere and materials, more energy would have to be used for the upwards portion than the downwards. So from the perspective of physics or geometry of spacetime, would it no longer be a 'perfect' circle? Is the upwards portion geometrically 'longer' (needing more force to maintain same velocity) and the downwards portion 'shorter'?
What I'm trying to get at is, would the forces created by uniform circular motions of a certain mass in different regions tell us something about - or even help us identify - the local shape of spacetime?
 A: 
What I'm trying to get at is, would the forces created by uniform circular motions of a certain mass in different regions tell us something about - or even help us identify - the local shape of spacetime?

A similar experiment is done at circular colliders, though the circles are horizontal and the geometry more complicated. In the need for great precision the earth tides have to be compensated.

The circular e+e− collider LEP located near Geneva is used to investigate the properties of the Z boson. The measurements of the Z boson mass and resonance width are of fundamental importance for the standard model of the electroweak interactions. They require a knowledge of the LEP beam energy with a precision of ∼ 20 ppm, which is provided by a measurement of the electron spin precession frequency. To extrapolate beam energy calibrations over a longer period of time, effects causing energy changes have to be taken into account. Among these are the terrestrial tides due to the sun and moon which move the Earth surface up and down. The lateral components of this motion modify the 26.7 km LEP circumference by about 1 mm. This change in length results in variations of the beam energy up to 120 ppm. 

What they have measured are gravity waves, and therefore  the change of the gravitational field of the earth.
In this sense the beams identify the local space time, if one would go to the trouble of transforming the Newtonian gravitational field into general relativity space time coordinates.
In your thought experiment, it would not be a perfect circle, unless the magnetic field (which you need to keep a particle in a circle) would correct for it, the same as with the beams at CERN. I would expect smaller velocity on the up side and higher on the down side , an ovoid.
A: A circle is defined in the following way:

The ratio of the perimeter $C=2\pi r$ and the diameter $d=2r$ is constant. This constant ratio is $\pi$.

Note that this definition holds only when Euclidean space i.e. flat space-time is considered. When we consider the curved space-time of general relativity, this definition doesn't hold as the space-time is no more Euclidean. When we consider a very small region of the curved space-time (called a locally-inertial region), we can approximate the region to be locally Euclidean.
However, note that a locally-inertial region is not exactly a flat geometry. The reason is that in such a region all the first derivatives of the metric tensor vanishes, whereas there are some non-vanishing second derivatives which determine the curvature of the space-time. So one can approximate an infinitesimally small region of the space-time to be locally flat (i.e. locally Euclidean) where the definition of the circle holds.
When we consider a larger region of the space-time, the geometry is no longer locally Euclidean and the effects of the curvature would emerge as there is a change in the local gravitational field.
A: There are a lot of factors on the earth like the air, gravity, and the elasticity of the materials. The motion described will be a perfect circle but the stress the object gets in is path will be different corresponding to the direction of the velocity vector and it's orientation to the gravity.
If for example let us assume the object is not freely moving but is connected to a shaft and is pivoted to one point. If the shaft is rotated the path described will be a perfect circle only at very low speeds, as the higher speeds are attained the structure will be elongated differently at different phases of the motion.
As for as the local shape of space time is concerned, I don't believe the change is so profound that it can affect the motion and the nature of it. And as far as the theory of relativity is concerned, it mainly is profound only at the speed of light.
The shape of the space time is only visibly noticed at very small distances and at very very large distances.
