Is the outcome of a fair coin flipped by a machine that is able to repeatedly flip a coin in an identical manner random?
Well when we say we have a fair coin then we don't only imply that the coin is fair because it has no bias to land on one side over the other, but we imply it is flipped with the same fairness so it remains random.
So normally your answer is yes.
But if for example the machine does not flip it randomly and gives a bias into one side then obviously the answer is no.
Yes.
Because i see a coin flipping machine as a procedure of flipping a coin and not as a mechanical thing.
Of course if we have to start worrying about how the machine will flip the coin then we go to chaotic dynamics, sensitivity on the initial conditions etc, but since we speak about mathematics identical(in the "flipping the coin in an identical manner" context) means 100% identical, so we have a deterministic procedure, so if we place each time the coin in exactly the same way all the times, on the machine, then obviously we would get the same side each time.
If we don't place it exactly the same on the machine, then we have to analyze how the way we place it affects the outcome, so there are multiple/countless cases to think about this.