When a new battery won’t charge, the culprit is often a defective charging cord/port or an electrical outlet that is not dispensing power. However, since you said that old battery swelled up large enough to crack the screen, I wonder if that battery didn’t do some internal damage as well?
If you’ve already tried all the software fixes (including updating the network card’s drivers), then the problem may be the card itself. If taking the card out of the slot and re-seating it doesn’t help, try replacing it altogether. Asus Vivobook Max X541SA-PD0703X Wi - Fi Card Replacement
When upgrading memory, it’s always safest to use brand/speed/type of RAM recommended on the motherboard’s manufacturer’s website. In a pinch, you can try the instructions provided in the following link (included in the instructions is a link to buy the correct RAM from Amazon): Lenovo Essential G560 RAM Replacement
Have you tried updating your Chromebook’s OS software to its latest version? If that doesn’t work, try employing the PowerWash function on your Chromebook to refresh the OS software completely. If that doesn’t work, try installing a USB mouse instead.
If none of the ports are working, the motherboard configuration may be the culprit. Are all the ports enabled in your computer’s BIOS/UEFI? If not, try enabling all the ports. Also, try downloading the latest BIOS/UEFI update from the motherboard’s manufacturer’s website. You can use the download to flash the BIOS and update the firmware.