Flash ADC negative values
Negative/positive is just relative to the point you choose to be zero.

In general ADCs can only work with input voltages within their supply voltage. For example an ADC powered from a 5 V source (ground and + 5 V) could accept input voltages between 0 and 5 V.

If we then choose 2.5 V to be the "zero" reference of the input signal anything below 2.5 V would represent a negative voltage, anything above 2.5 V a positive voltage.

But what if I have a sinewave between -1 V and + 1 V ?

Then you can either "level shift" it, I mean add 2.5 V to it. For an audio signal where we're not interested in the DC value, this can be done by AC coupling with a capacitor and some resistors.

Or (more practical for situations where you do want the DC value) you can power the ADC with +2.5 V and -2.5 V.

So basically you "see" a problem where there isn't one. Indeed ADCs by themselves usually cannot work with negative values but there is no need. We can just "shift" the ADC's working point so that all input voltages become positive. This "shifting" of the input voltage is quite trivial.