Battery Power Management PCB Design
I'd suggest you go to TIs website (or to other similar manufacturers) and look for battery charging ICs. They normally have typical application circuits that could help you understand what needs to be done and how you could achieve this.

The same for the PoE ports. There are normally specialized regulators for this application.

Additionally study application notes that normally accompany the ICs.

Also study the existing circuit (of course only if a schematic is available, otherwise it could be challenging) and try to understand how it was implemented.

I would say more or less you need the following:
- A conversion of the input power source to a voltage that can be used on the board (for example an AC to DC converter and/or further step-down converters).
- Then battery charging ICs that take care of the actual charging process.
- PoE regulators that provide power to the PoE ports.