So, I have a LiPo battery with protection module. Can I just connect the LiPo and a LOAD in parallel to (Charger A) B+ B-, in order to do the same as (Charger B) (LiPo to B+B-) (LOAD to OUT+OUT-)? I think not? Here, let me call the diff. between (Charger A) and (Charger B) ( the DW01A and FS8205A part) a "extra circuit". Since my LiPo already have protection module. Can I skip parts of the extra circuit? Or do I need to add full part of the extra circuit? To build a (Charger B) from (Charger A)?
As I interpret the question, it is about how to make a low-voltage uninterruptible power supply. You can't simply connect the charger and battery to your load even if your battery has DW01A protection circuit. You will need a "power path switch", which will disconnect the battery from load in case if the external 5V power is permanently present, and switch to battery power only if the external source is lost.
The reason is that your battery will be overcharged when you will keep the charger feeding both your load and charging your battery, the battery will be overexposed to charging voltage.
There are two kinds of overcharge: overvoltage (and DW01 will protect your battery from that), and overexpose to nominal charge voltage, from which the DW01 won't protect, it doesn't have the charge threshold protection. Charge termination is a function of TP4056, but since your load will still consume some current, the termination condition might never occur, and TP4056 will think that the charge needs to be continued. This will quickly damage your cell.
That's why all reputable battery-powered systems employ extra circuitry called "power-path switch", which is usually embedded into charger, or there are stand-alone ICs with this function. An example would be this simplified diagram, from Texas Instruments:
As one can see, the battery is not connected directly to system load, which allows the charging section of IC to complete the standard Li-Ion charging cycle and disconnect when battery is full. And then charge it in trickle mode if external source is permanently there.