Reading Flow Meters from Mic Port
You have a lot of questions in this post. Next time, I recommend splitting them into a separate posts, or at least numbering them.

1) Yes, you can get 5V from charging USB cable. This will work just fine.

2) Yes, two-resistor approach you are describing is good, as long as you use it with right kind of output (push-pull or open collector). The schematics you have shown assume push-pull outputs, while FT-330 is open collector (you can tell because they require pull-up resistor). It is not clear what kind of output adafruit meter has, but presumably push-pull, since there is no mention of pull-up resistors.

3) I am not sure why you need op-amp at all. It will not hurt, but you should be able to just connect output to microphone's input. The output voltage will drop a bit, depending on phone's input resistance, but this may not be a problem. Only add op-amp once you have established the phone provides too much of a load.

4) The reason your op-amps do not work is because you made an wiring error. From your schematics, your Vcc should always be 5 volt. If you do not see it, triple-check you have assembled everything properly.

extra: from your screenshot, it is clear you have a capacitor on your microphone input (likely inside the phone). As you can see, this has an effect that an 'average' voltage one the input will be 0 volt. You can see it on the graph -- if you give, say +5V volts, then in few tens of ms, the capacitor will charge to +5V and the input will see 0 volts(+5 - +5). If you now give 0V on input, the card will see -5 volts (0 - +5V). But soon, capacitor will discharge and you will see 0V again.