Ok. I notice several things about your schematic. Some are important, some are not. You did not draw an active filter on the first stage. That is fine, we can add it later. Let’s just get the main functions working first. R3 needs to be omitted and the non-inverting input (+) on the second op-amp needs to be tied to about 8.5 volts instead of ground. This can be accomplished with a voltage divider. I assume you know how to make a voltage divider out of 2 resistors tied between the 9 volt supply and ground. If not, just ask. When I say ground, I mean the negative side of the 9 volt supply. The negative side of the mic is also tied to what I call ground. Also, tie the non-inverting input of the first op-amp to a 4.5 volt reference. Use a voltage divider to get it.
Now, to explain why this circuit does what I say. When the non-inverting input on the second stage is tied to 8.5 volts the output will also be centered around 8.5 volts. Since the cap is on the input the DC gain of the stage is one. When you couple an AC signal in through the cap the output should swing positive and negative by an amount determined by the feedback resistors. However, it cannot swing farther up than the 9 volt supply so it will mostly swing down. And obviously it cannot swing lower than ground. When there is no AC signal coming in then the output will be sitting at about 8.5 volts. This is just what we want. With adjusting the gain we can set the circuit for just the amount of noise we want to trigger the timer.
You also drew the output of the second op-amp going directly into the 555. We are going to change this. Insert a diode in between the op-amp output and pin 2 of the 555 with the cathode towards the 555. Tie a resistor from pin 2 of the 555 to ground.
Next we are going to make the manual reset circuit. We’ll make a sort of Schmitt trigger circuit out of one of the op-amps left on the quad package as a type of latch. Tie a resistor from the non-inverting input to the output. Tie another resistor from the non-inverting input to ground. This is positive feedback that makes a Schmitt trigger always go to either power supply rail. This is just what we want. Tie the inverting input to a 4.5 volt reference made out another voltage divider. Tie the output of this op-amp to pin 2 of the 555 with a diode inserted in series. Cathode towards the 555. Insert a diode with the cathode on non-inverting input on the op-amp and the anode on the output of the 555 timer Also insert a resistor in series with this diode. Now install a momentary switch from the non-inverting input to ground. This is the reset button. Also tie a .47 microfarad capacitor in parallel with the switch.
Draw the schematic, ask any questions and I’ll be glad to help you.