Each section is a non-inverting amplifier with low-pass filtering. The cut-off frequency is set by R2-C2 and R8-C4, respectively:
$$
f_c=\frac{1}{2\pi R C}
$$
But also each section has high-pass filtering at their inputs, so combining with low-pass filtering they form a non-inverting amplifier with band-pass filtering.
Now the answers:
What is the purpose of applying a voltage to the inverting port of the OP amp via R4, D1, and D2. I have been told that the two diodes are used to lift up the potential to ensure the arterial pulse can go through completely. I am having trouble understanding this.
The network bring an offset so the reference point of the amplifiers to be 1.2V instead of 0V. The main purpose is to prevent signal clipping. Since the op-amps have no negative supply, the amplified signal may hit the GND which means clipping. With adding an offset of approx 1.2V the risk of clipping can be eliminated (depending on the amplitude of the input signal).
What is the purpose of the potentiometer R7 in this circuit?
Gain adjustment without changing the pass band. Normally the gain could be adjusted via either R2 or R8 but since they interact with the parallel caps the pass band would change. So R7 basically forms a voltage divider and scales down the output of first stage. The voltage fed to the second stage can be lower so the output can be adjusted. The maximum output will be achieved when the wiper of R7 is at top-end.
Why are there high pass and low pass filters for both step? The band pass filters I search online only uses one for each step. Is it supposed to act like a two-step narrowing process?
I don't know the corner frequencies as some of the component values are not given. But the main purpose of making two-stage filtering is just to have "harsher" filtering. You know the filters are not behave like a brick-wall. Some of the frequencies beyond pass-band will still appear at the output but with lower amplitude (the bode diagram will tell you). With two-stage filtering having the same corner frequencies the amplitudes of these unwanted frequencies will be lower. Therefore the higher the number of stages the more close to brick-wall filtering.
How are the gains of the two op amps correlate? Should they be the same?
They don't have to. Depends on the design and requirements. The first stage can have relatively higher gain as the output can be scaled down with R7.
