how does the SDR sample rate translate to sample rate and bit depth of the resulting audio signal?
Extremely indirectly. I haven't got precise mathematical formulas for you, but here's an overview of the topic which should let you figure out where you want to dig deeper.
First of all, FM is an analog modulation. This means that there is no inherent sample rate or bit depth of the signal recovered by demodulating, but only an audio bandwidth and noise floor. So let's look at what influences that:
The bits per sample (bit depth) of the digitized RF determines the amount of error introduced by the quantization (conversion to discrete samples), which can be described as quantization noise.
Note that quantization noise is at a constant level in the digital signal coming from the ADC. This means that — unlike radio interference — the signal-to-quantization-noise ratio improves the more you turn up your receiver's gain. But you can only do that so far before you get clipping (signal level exceeding the representable sample values).
Therefore, the best results are obtained when you have a narrow analog filter in your receiver to remove all but the desired signal before digitization — the wider bandwidth you receive, the more total power is arriving, and so the less gain you can use before clipping.
Of course, using a narrow bandwidth prevents you from having those pretty waterfall displays of ten stations at once. It's a tradeoff.
For more words on the subject, see my previous answer over on Amateur Radio Stack Exchange, to the question Why is dynamic range relevant for an SDR?
A simple analysis would suggest that the sample rate of the digitized RF has no effect, because it's just bringing in a wider bandwidth, full of unrelated signals and noise, than we actually need. However, this is not true, if the ADC has fewer bits per sample than the actual DSP processing (for example, 8-bit integer samples from the ADC going into 32-bit floating-point DSP operations). The extra samples contain some of the information that was lost in quantization; resampling the unnecessarily-high-rate signal to a lower rate will automatically use that information, so those 32 bits aren't actually wasted.
So to combine this with the previous point, the best case given fixed hardware is to configure it (if possible) for the narrowest analog filter bandwidth and the highest sample rate (creating a signal whose spectrum has an obvious filter hump in the middle) and then resampling it before demodulation.
To sum up, the sample rate and bit depth of the incoming RF determines how much noise is added to the signal by the process of digitization. The next question is how that noise, the RF noise entering the software demodulator, affects the noise level of the resulting audio, and I'm afraid I don't know that and haven't found an obvious reference.
Now that we know that our input can be considered as a signal with more or less noise depending on its parameters, let's look at what to do with the output.
As Nyquist tells us, the sample rate of the output audio signal does not need to be greater than twice the highest frequency present in the signal. That's the audio bandwidth, which for broadcast FM is, let's say, 15 kHz. Therefore, the required sample rate is 30 kHz, ideally. However, the signal must be filtered to remove noise that is outside the expected bandwidth, and using exactly 30 kHz sample rate means that the noise which passes the necessarily-imperfect filter will be aliased down into lower frequencies. Thus, it is better to use a higher sample rate to allow some room for the filter transition band — perhaps 44.1 or 48 kHz, because those are common audio sample rates.
Quantization introduces quantization noise — but we already have noise in our signal. Therefore, the bit depth of the audio should be chosen so that it introduces an insignificant amount of quantization noise compared to the noise already present. (Note: You can't avoid introducing any noise at all. Your finite-precision arithmetic is doing that at every step.)