> You can make an observation of a system whose state is undetermined. By interrogating the system for its state, a state becomes determined.For QM, this is not correct, although it's a common misstatement. The correct statement is this: you can make an observation of a system which is not in an eigenstate of the measurement operator you are using. After the measurement, the system is now in an eigenstate of the measurement operator--i.e., the act of measurement changes the state.
Note that this is only true on a collapse interpretation, like Copenhagen. On a no-collapse interpretation, like MWI, the "observation" is just an interaction that entangles the state of the measuring device with the state of the system being measured--it's all just unitary evolution.
> You could flip the coin, and put its state vector into the form of sqrt(2)/2 Heads + sqrt(2)/2 * Tails. This state isn't observable*
Yes, it is; but it isn't observable by a simple method like looking to see if the coin is heads or tails. But according to QM, every state is an eigenstate of some operator, so there will be some observation that will distinguish sqrt(2)/2 * Heads + sqrt(2)/2 * Tails from the state that is exactly orthogonal to it, which is sqrt(2)/2 * Heads - sqrt(2)/2 * Tails.