AT&T technician here 👋 You have the right idea from taking the ethernet cable from the ONT (white box) and plugging it into the red port on your gateway (I’m assuming it’s a BGW210). After this you have two options, you can either put RJ45 mail ends on the ethernet cables and plug them directly into the gateway, or you can buy a patch panel and use a punch down tool to punch the wires down into the patch panel.
Due to the fact that you have 15 ethernet ports, and there’s only four on the BGW210 gateway, You will need to buy a gigabit Ethernet switch if you want to use more than 4 ethernet ports which is what the gateway only allows. Since your panel has electrical outlets at the bottom this will be no problem.
I’m about 98% sure that you will need to use the T-568B wiring diagram (diagram can be found online, make sure to use B). However if you want to confirm this before you do 15 jacks, you can unscrew the faceplate on the ethernet jacks in the room and verify if they are T-568A or T-568B. Again, because this is an apartment I’m assuming that there are ethernet jacks in the apartment and you just need to complete the other ends which is located inside of this panel.
That’s how cellular Internet works. It can be extremely good in 1 area and absolutely horrible in another area. It all depends on the location of the tower as well as the load on the tower. Get another internet service provider would be my answer. Ask your neighbors what they have or look up online through the big name providers (AT&T, Spectrum, etc)