In the case with a 1-D input like audio, how can a system with one input do many different kinds of processing at different sampling rates and still, at the end of the processing, produce one waveform? For example an ADC/DAC reading and writing at 48kHz, but has an FFT running at 8kHz or another set of processes running at 16 kHz? Is it a matter of decimating and interpolating? How would all of these processes line up?
You are on the right track. You can use decimation and interpolation (or resampling, which is a combination of both decimation and interpolation) to change the sample rate. For instance, if you start with 48 kHz samples you can decimate by 6 to get down to 8 kHz, do some processing, then interpolate by 2 to get to 16 kHz, do some processing, then interpolate by 3 to get back to 48 kHz.
"Lining up the processes" is not in and of itself difficult. You can decimate and interpolate whenever you want. Doing it in a way that makes sense and is useful, on the other hand, requires knowing more about what you start out with and what you want to accomplish.
If you are running a causal/real-time process, note that any upsampling/downsampling sub-process involves anti-alias filtering that will add a delay (a linear delay in the case of a symmetric FIR filter).
So, if you want to re-combine resampled and non-resampled processing chains, you might want to add delays to the non-resampled processing paths to match the anti-alias filter (and other) delays in the resampled processing paths to "line things up", depending on what you are doing, of course.