why is static friction acting radially inward and thus providing centripetal acceleration?
Because the disk underneath the coin is being accelerated radially inward.
The linear case may be easier to understand. We place a package on a conveyor belt. The belt accelerates to the right. The frictional forces between the package and the belt accelerate the package to the right.
The turntable is a rigid object. Whatever is applying torque to the table causes the entire object to spin. The rigid cohesive forces mean that parts of the turntable at some distance from the axis are accelerating toward the center. Just like the conveyor belt, friction attempts to accelerate the coin to the same extent. Either the coin rotates with the turntable, or friction is insufficient and the coin moves in some other path.
what causes the the coin to obtain a velocity in the first place (i.e. what causes tangential acceleration if the static friction acts radially inward)?
The radial acceleration is for the constant rotational speed case. If the turntable is changing speeds (starting from rest), then there will be tangential acceleration as well.
what is causing the centrifugal force required to increase making the static friction insufficient?
Required centrifugal force depends on the mass of the coin, the distance from the axis, and the rotational speed of the table. If all of them are constant, you would expect the required force to remain constant and the coin to never move (relative to the turntable).
In practice, the table might not be perfectly flat. A bump might disturb the coin causing friction to reduce. Or maybe the motor changed speed for a second. The conditions of the experiment would determine if the coin were to stay or not.
i thought it was the relative velocity and not the acceleration that determined friction.
That's correct (at least for kinetic friction). But assuming static friction in this case, the relative velocity is always zero. So the force of friction is sufficient to accelerate the coin exactly as much as the turntable underneath is accelerating (up until it begins slipping). But if the turnable underneath were not accelerating, then the coin would not require friction.
The coin would still speed up if i placrd it on a moving disk; whats providing the tangential acceleration then?
Kinetic friction. It is only in the case where the coin has no relative velocity, and that the turntable is not changing the rotational speed that friction is purely radial. If you are in a slipping configuration, then friction will act in other directions as well.
