(Beginner) Hacking TCM-20DV sony walkman speed control
First question: why would the first resistor take the whole drop?


Because the motor has a very small resistance.


  Second question: why would it need physical nudging to get started in this case?


Probably the series resistor didn't allow enough current to flow for the motor to produce sufficient torque to overcome static friction.


  And third question: what is the appropriate approach to this problem?


Connect the motor to the battery with just an ammeter in series and see how much current flows. Gently apply some braking to the shaft with your finger and note how much current flows.


  And any reference material you think is relevant would be much appreciated!


Recommending specific sources in answers is discouraged here. There is a lot of material on the internet. Whatever you can best understand is best for you. It is best to start by studying basic principals and then move on to the sort of thing you are trying to do.

The basic understanding that you seem to be lacking is that a voltage divider must always take into account the connected load. The motor is, in effect, a low resistance load. You can begin to understand that by measuring how much current the motor takes when loaded and unloaded. The motor is best understood as a very small resistor in series with a generator that produces a "back EMF" voltage that opposes the current from the supply and varies in proportion to speed. When the motor is not turning, back EMF is zero. At full speed, back EMF is near the supply voltage.

Controlling a very small motor with a variable resistance in series is fine for a beginner. Historically, that is how DC motors were first controlled. However that is very inefficient and does not provide very accurate control. You need to pay close attention to the power rating of the series resistor.

Someone else might recommend some reference material, but I believe you should learn to find on your own, material that suits your level of experience and style of learning.