It seems that the paper "Measure theory through dynamical eyes" by Katok and Climenhaga here (see, for example, Section 1.4) will be useful (1). I hope this helps.

It seems that the nice set of notes "Measure theory through dynamical eyes" by Katok and Climenhaga here (see, for example, Section 1.4) will be useful for the first question (1). It is also mentioned that if the ergodic system has more than one orbit, then ergodicity is sufficient for nonmeasurability of the partition (2). I hope this is helpful. Perhaps Vaughan Climenhaga will see this and write a nice answer.