How can I use a DFPlayer Mini to play speech and control a servo to move an animatronic mouth based on audio amplitude?

Ok so that's not so bad. Don't overcomplicate it more than it needs to be.

Mouth can just move randomly whenever one of the responses is being played. No-one says it has to realistically mimic human speech patterns, which would be very tricky.

You will have to identify the direction of the person speaking. You could then simply:

  1. move eyes first in the direction of the speaker
  2. rotate head in the direction of speaker
  3. once head has reached end of travel, straighten eyes again

That would meet the requirements.

It sounds like the speech recognition will be the most sophisticated part. The pre-recorded responses would not be hard and could be done in one of the Arduinos that can play audio from an SD card, as I suggested in your other thread. No need for external MP3 players or suchlike.