One could argue even classical mechanics isn't deterministic as we think of it because of chaos, which has fascinating connections with QM. W Hoover (of the Nose-Hoover thermostat fame) did some great work with reversible thermostats exploring the instability of Newtons equations of motion.

Measurement is a transfer of a state information from a subsystem 'being measured' to a 'measurement device' caused by an interaction. (ie: a quantum bus is a measurement) Just like in a classical situation, further dynamics can depend on the final state in the measurement device. The only difference is that all measurement states will be involved in the quantum measurement device, and all possibilities of dynamics conditioned on the measurement are explored. At any point in the future the whole chain of events can be collapsed, if the measurement device interacts with a classical incoherent object.

That's correct, But there are numerous good methods to propogate irreversible dynamics in qm. Many of these are exact in the limit of a noninteracting bath of bilinearly coupled oscillators which is sufficient to describe a measurement collapse. There's no mystery or inexact proscription to such a simulation of a collapse process. It's just complicated.