PIR Detector Controller IC
References

Background: http://www.digikey.com/us/en/techzone/sensors/resources/articles/design-considerations-for-passive-infrared-sensors.html

More theory: http://media.digikey.com/pdf/Data%20Sheets/Cypress%20PDFs/CY3236A-PIRMOTION%20PIR,EVK.pdf

It looks like this is the sort of sensor: http://pdf1.alldatasheet.com/datasheet-pdf/view/191226/MURATA/IRA-E700.html

Answers


Changing the oscillator frequency will affect all the timings based on it, making them longer or shorter. For example going from 62.5Hz to 50Hz will make all the times about 20% longer. I would avoid changing it very far; it will give you worse results and the datasheet doesn't say what the max and min are.
Supplying your own voltage to the sensor will not change the references used by the comparator windows. My guess is that this would change the center of the temperature range. Fortunately, this isn't necessary: the IRA-E700 example sensor claims a voltage range from 2 to 15V, which is nice.

The sensor isn't a digital device with a specific level, it's a lot more like an electret microphone: a sensing element connected directly to a MOSFET which amplifies its tiny signal.

The reason why you get a positive and negative pulse is that there are two sense elements in the sensor, connected in opposite directions. One produces positive pulses when it "sees" human body warmth, the other produces negative pulses. So you have to move from one to the other to trigger it, but it won't trigger on changes in ambient temperature that affect both.
A push-pull output is an output that works in both directions, unlike e.g. open collector outputs which can only sink current. The \$V_{oh}\$ and \$V_{ol}\$ values are its output high and low voltage specifications.
R4: this sets the voltage gain in the pair of amplifiers. The signal from the sensor is tiny, and amplified quite a lot. Just how much is set by R4.


Generally it's much easier to buy a PIR module with integrated sensor and detector than try to combine the two yourself using this chip and some unspecified sensor.