When using an op-amp, inverting amplifier is this formula : Vout = - ( Rf / Ri ) * Vin
For non-inverting amplifier : Vout = ( 1+( Rf / Ri ) ) * Vin
Here a site to help you understand op-amps. http://www.play-hookey.com/analog/
Op-Amp operated at dual voltage : + side / gnd / - side ex: +5 . gnd . - 5
Bear in mind that the Arduino don't "like" negative at an analog pin, it will damage the ATmega chip
Is the Drain of the PIR the OUTPUT of it, Source would be the voltage supply (5V), Ground would be the Ground.
YES . GND is GND. Drain is OUTPUT signal. NO Source is +5 V. A resistor between Source & GND. A resistor between +5 V and Drain.
Look under JFET transistor use as an amplifier. http://www.electronics-tutorials.ws/amplifier/amp_3.html
You will know why.
If you have access to an osciloscope, to see what signal come out of the sensor, than the need to design a circuit to convert the signal into a TTL signal, if it is needed.
This is base on the datasheet of the sensor.