Understanding AFE4403 datasheet
The datasheet is very long and I found it hard to grasp the essential
information about the AFE4403 (https://www.ti.com/product/AFE4403).

It does a lot of things. Just take your time and work through it steadily. The page you linked to also has a number of other documents to help you use the AFE4403. You can study them as well to help understand how to use this ic. This page discusses the principles behind pulse oximetry.

What are the functional differences exactly between the Microcontroller, AFE4403 and the PulseOx Connectors? In other words,
do we really need all of them? Why can't we connect the DB-9 to the
microcontroller without the need for an AFE4403 (or the LEDs and a
photodiode to the microcontroller)?

I can't see any mention of DB-9 connectors (or, correctly DE-9 connectors) in the application note. You don't have to have a connector. There are plenty of self-contained pulse oximeters with no external connection.
AFE in AFE4403 stands for analogue front end. The red and infra-red illumination LEDs need to be switched with precise current control and a very weak signal is returned from the receiving photodiode. You can't just connect them up to GPIO pins on a microcontroller board because they are not compatible. You don't have to use an AFE4403 but you would have to provide your own precision analogue front end.

It seems that we can connect 3 LEDs via a common anode. If the LEDs are identical (say infrared with forward voltage and current 3.3V
and 1 A, respectively), what is the maximum number of LEDs that we can
connect in parallel such that they are driven by the AFE4403
simultaneously?

Table 7.7 (Electrical Characteristics - Transmitter) on page 7 of the data sheet tells you the maximum LED current is 100 mA. Figures 56 and 57 on page 41 tell you a maximum of 100 mA can flow through LED1 or LED2 (alternately, not both simultaneously) or you can sink 50 mA through TX3. So you can connect one-tenth of your preferred diode. Note the requirement to match forward voltages on page 43 if you do want to parallel connect diodes (within the 100 mA limit).
As a general tip when reading tables in data sheets make sure you pay attention to the test conditions and any footnotes because you'll need to recreate those conditions to achieve the documented performance.

how to switch between the H-bridge configuration and the common anode configuration (pages 41 and 42)? Do we need to have separate
PCBs, one for each configuration?

The application note implies you choose one configuration so separate PCBs for different uses would be easier. Note that operation depends on precision control of the LED current so if you wanted switchable configuration you'd need to make sure you didn't interfere with the current control.
Edit: Follow-up questions answered here for more space.

In page 42 of the datasheet, a maximum of 100mA current can flow
through an LED (H-bridge). 1) Looking at page 22 of the development
guide (ti.com/lit/ug/slau572b/…), there are two currents for each LED
[i.e. Current in mA, and current setting]. What is the difference
between them, and is the first current (1.96078mA) the actual current
flowing through the LED?

Yes 1.96078mA is the current flowing through the LED. The current setting (20) is the multiplier for the 8-bit resolution current control.
See Fig 52 (data sheet p.38). You choose a current reference for either LED which sets the maximum current. The chosen maximum current reference is then subject to an 8-bit resolution current control, i.e. the chosen maximum is multiplied by (0-&gt;255)/255.
Table 7 (data sheet p.74) shows the full-scale LED current reference can be set to 0 (off), 25, 50, 75 or 100 mA. The following equations 7 &amp; 8 show how the current is calculated (though the denominator needs to be 255 rather than 256).
While I don't have the application software I assume the drop-down box labelled LED CURRENT CONTROL DAC in fig 24 (development guide p22) lets you choose from the 0, 25, 50, 75, 100 mA options of table 7. The LED n Current Setting box should be the multiplier sent to the 8-bit LED Current Control multiplier. So the LED output current should be 25/255 x 20 mA or 1.96078 mA.

If the values of currents are set to be different for each of the two
LEDs (e.g. 1mA and 4mA), why the intensity of the two green LEDs is
the same?

Measurement of oxygen saturation in blood depends on measuring the ratio of red to infra-red light transmission hence the careful control and measurement of the red and IR light signals. Measurement of pulse rate doesn't need this differential signal so the drive requirements are easier, e.g. section 9.2.1 (data sheet p.83).