To comply with Commendation Criterion 34, does the continuous professional development for our CME team have to be directly related to CME? What about planning/managing trainings?

Not necessarily.  If your CME team includes individuals who are meeting planning staff, then it would be compliant to include education/training for them about how to be more effective meeting planners.  This would effectively contribute towards the expectation of continuous professional development for Criterion 34.