As far as I know, the 'DSP' is an 8051 or compatible microcontroller, which is connected to a D/A converter. Then the output is fed into an analog mixer.
On the SB Pro and newer cards, this mixer has digital volume control.
There may be various levels of integration of the circuits.
Anyway, the way the PCM audio works on a Sound Blaster is like this:
The 8051 has an internal timer, and is programmed with a divider value to select the actual sample rate (this is how the SB can basically get 'any' sample rate between 4 kHz and 44.1 kHz).
The 8051 will fetch one new sample at every timer/divider interval, and will send it to the D/A converter. So the D/A covnerter is a really basic circuit (like a resistor ladder), and does not have any kind of clock or timing on its own. It just latches a single value, and will continue to output it until a new value is written to it.
Very basic analog low-pass filtering is applied to reduce the aliasing caused by this method. The SB16 introduced a more advanced low-pass filter that was adaptive to the chosen sample rate.