LM358 is quite noisy, but if the microphone provides enough signal (>10 mV) it might be adequate. TL071 / TL072 are much better opamps, but need a higher voltage supply, like 12 V and the output should be capacitively coupled to the ADC pin, and the pin biased to VCC/2 with resistors.
The problem with the schematic you posted is the low input impedance, 1k -- because of the inverting configuration, thus loading the output impedance of the microphone.
Use the non-inverting configuration, with higher value input resistors, like in this schematic. Capacitors C3 and C4 provide a bit of low pass filtering.
Also, you may try to reduce electret microphone bias resistor to 4k7, or around this value.
You can't connect directly a low impedance speaker to a low power opamp like LM358, because its current capability is very low.
With the schematic posted you should only use LM358 and a 5V supply, with a higher voltage you may destroy the ADC.
You can eliminate the output capacitor, C5, and only use the To_ADC output to connect to the ADC or probe with the oscilloscope in DC coupling mode. I had the impression you needed a secondary AC coupled output.
If you want to use TL071 with a 9V supply, it should be also biased at half voltage, 4.5V, to function correctly. Then the output should be AC coupled to the ADC input, that is separately biased at 2.5 V, using resistors R6 and R7. I recommend you don't eliminate R5, because it will reduce the inrush current applied to the ADC input (hopefully the ADC has internal clamping diodes) at power-up when C5 is discharged.
Don't guess, look on its datasheet. It is too old to be rail-to-rail.I am guessing TL071 is rail-to-rail
The same as any other non-inverting opamp circuit. The gain is (R2/R1) +1= 101 times. R2 and R1 provide negative feedback.I am curious how gain is set on the TL071 circuit you proposed.
Some of the schematics you found WILL NOT WORK! The first one produces only a DC pulse when there is a sound.Currently the plan is to use 5V supply on my initial LM358 design I found online as posted above and bias the output at 2.5V to see how it performs. Hopefully I can get it to swing decently as long as it doesn't manage to get past 5V and blow the ADC.
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