Please review my schematic design
68 Ohm resistor before LM386 because the maximum recommended input
  voltage is 400mV. When all bits are high, the output voltage is
  ~3.10V. Output impedance of R-2R DAC is equal to R. Using a voltage
  divider, 3.10(68/510+68) = ~360mV.


The LM386 is supposed to work with AC input. Your DAC output is DC, so you will only get half the expected output voltage swing. You should AC couple the signal to the LM386.

With 4 bits you only have 16 steps, which will be very audible. You can reduce stepping noise by adding a low pass filter. Since the DAC produces 3.1Vpp but you only need ~360mV, the filter can be a simple passive type with high insertion loss.

Something like this:-







simulate this circuit &ndash; Schematic created using CircuitLab