MAX9814 feedback from headphones - normal, or faulty device?
The MAX9814 has an automatic gain control - it jacks up the gain to try to get a constant output level. It also has several selectable gain settings - the highest gain is 60dB

The MAX4466 doesn't have automatic gain.  It has fixed (or manually variable) gain.  It is a simple op-amp that can be used for other purposes.

What you are hearing is feedback - the microphone picks up a sound, the amplifier makes it louder, and sends out back out through the speaker.  The microphone hears the output from the speaker and makes it louder.  Round and round it goes, getting louder all the time.  This feedback loop is same thing that makes an oscillator oscillate - and that's what your amplifier does.  It oscillates at a frequency that depends on the time it takes the signal to make one round trip - and since that depends on the distance from microphone to speaker, you get a tone whose frequency gets lower as the distance from microphone to speaker grows.

This happens with the MAX9814 and not the MAX4466 because the circuit with the MAX9814 amplifies more (has a higher gain) than the circuit with the MAX4466.

To stop the feedback squeal, you can select a lower gain on the MAX9814.  Download the datasheet, and have a look at it.  It tells you how to set the gain.  Try it.  See how far you get.  If you have trouble, come back and ask for help changing the gain (new question, reference this one if needed.)



This has nothing to do with RF or the doppler effect.

You have no transmitter or receiver, so no RF. 

The sound changes with distance, not motion so it is not the doppler effect.



And finally:  

Yes, this is what you should expect when connecting a microphone to a speaker through a high gain amplifier.