LM35 and 4-20 mA output

The gain of the circuit is +11 (Gain = 1 + \$\frac{R_F}{R_1}\$). This means when the input sinewave is positive 0.1V you'd expect an output that is +1.1V and when the input is negative 0.1V you'd expect an output that is -1.1V. These numbers are all relative to ground. The first problem is that your op-amp has supplies that are +9V and 0V so immediately it's clear that the op-amp can't produce the -1.1V or indeed any negative voltages.

What to do? Both the input signal and the ground connection of \$R_1\$ need to be set to halfway between 9V and 0V - at the moment you're throwing darts at a board that is 2 foot higher than you thought - you are just hitting "three" when you should be aiming for bullseye (sorry about the darts analogy but it seems apt!).

OK, so the problem is that your input AC signal is 0V (or ground) referenced - if you just wish to amplify AC signals then you can use a capacitor (that blocks the steady 0V connection) and then have a resistor divider that sets midrail on the input at 4.5V. You have to do the same for \$R_1\$ - I would make R1 2 x 200ohm resistors; one going to +9V and one going to 0V and joined together at the -Vin input.

That's a start; then get a better op-amp that can deal with rail-to-rail outputs - you might achieve 8.8Vp-p from this circuit with a 0.8Vp-p input.

This is basically a classic window comparator, with stuff around it to make is actually useful in the particular application.

PIR sensors report changes in IR accross the sensor area. C2 removes the DC bias, and the circuit around IC1D amplifies the result and also does some frequency filtering. This is probably in part to reduce frequencies that aren't relevant and therefore just add noise, and in part to get the response of the overall sensor+filter that is useful for detecting motion.

IC1C and IC1B are the window comparator. R7, R8, R9, and R10 are a divider chain making voltages for the output of IC1D to be compared against. Just from the topology without looking at any numbers, you can see that the threshold for IC1C is higher than that for IC1B. Also see that the input signal into the window comparator (outout of IC1D) is fed in to the two comparators (IC1C and IC1B) at opposite polarity. In the "window" region, which is the voltage range between the - input of IC1C and the + input of IC1B, both amps will be driving low. Below the window region, IC1B will drive high and IC1C low. Above the window, IC1C will drive high and IC1B low.

The two comparator outputs are averaged by R11 and R12, then the result compared to a threshold by IC1A. This threshold is set so that IC1A drives high only when both comparator amps are driving low, meaning the voltage is in the window region.

The digital signal that indicates whether the sensor output is within the window region is capacitively coupled into this HT2812 thing. I didn't look that up, so I don't know what exactly it does, but from the transistor and speaker it is probably intended to produce a beep when motion is detected.

I'm not sure what the point of the switch in series with the KEY input is. When the switch is open, the HT2812 block won't receive the motion signal. If that is the intent, then powering everything down would be the more obvious approach, so there is probably some additional feature it supports. I don't know why you'd want to only sound a beep due to motion when a button is pressed, but that appears to be what what this circuit will do.