...she has_had/has_got her moles removed by the doctor
Forms of Have or get are both possible in a lot of causative structures. I think has/had is more suitable here.
she had/has her moles taken away by...
Taken away probably wouldn't be used in this context.
she removed/took_away her moles
Again, removed would be the common choice. However I don't think removing a mole is something one would want to do oneself! Ouch! You should use your first causative construction to indicate that someone performed the action for her; either with or without the "by" phrase which indicates who did it (the agent). Such an addition is optional and especially when speaking of medical procedures as the identification of the agent is normally obvious (a doctor/medical professional).
How does a native say such a thing, please?
Student or Learner