It is only treated this way statistically for many practical purposes. Nothing in our universe is inconsistent with a purely deterministic model, and certain properties of the universe are only expressed in deterministic systems which lends some credence to the concept. Quantum phenomenon in particular have been formulated as expressions of deterministic processes (whether those specific formulations map to reality is unknown -- they only prove the possibility). Papers have been published on this.
There are many classes of simple finite state systems that cannot be perceived as anything but random even if you had an intelligent machine with the full state space of a finite universe at your disposal. For example, strong cryptography is premised on this fact and uses algorithms with exactly this property.
Solomonoff induction is one of the most brutally limiting concepts in mathematics, and somewhat analogous to the incompleteness theorem but in systems theory. There are a great many things about any finite state system that can never be known from within that same system. Quantum phenomena my very well fall under this umbrella such that even if we can know that it is deterministic in fact, we can never treat it as such as a practical matter because we cannot measure the state of any particular instance.
Demonstrating that a process is finite state to extremely high certainty is cheap and trivial. Determining the actual state of the same process is typically intractable.
No, that's not correct. The state evolves determinstically according to the theory but the state is not the observable. The observable phenomena are "generated" from the state in a random manner, again according to the theory. It is not a matter of practicality - there is currently no better description.