Router boxes and equipment

If you were installed today, you'd get one box to replace all three of those, and a wall plate inside the house that a fiber jumper would connect to from your Gateway (a BGW320, replacing your 5268AC).

As it is, the power supply with build-in UPS, the external ONT and your Gateway are all in use.

Just to be sure that the inside box is still in use, the white wire coming out of the outside box goes up past the dead meter box, around the soffit, down the outside wall, then through the wall into the space below the sink. It then goes under the house crawl space and up into the black router box in the office.

The cord that comes out of the bottom of the outside box goes through the wall into the white inside box which is plugged into the duplex outlet.

I noticed that the inside box does not have a battery in it. If it is still in use, is that recommended as backup? Thanks for your help!

First picture, near roof is an NID, which would have been used for POTS and/or DSL back in the day, and unless you still have POTS, is no longer used.  The cable going from the ONT (bottom of first picture, center of third) past the old meter box, past the NID to the 5268AC, is a twisted pair cable carrying your Internet service Ethernet signal from your ONT to your 5268AC.

The cable in the second picture that goes to the box in the fourth picture carries power from the box in the fourth picture to your ONT (which is inside the box in the third picture). 

The box in the last picture is a power supply for the ONT.  If it doesn't have a battery, your ONT will not work during a power failure.  OTOH, if your 5268AC doesn't have an installed backup battery, or isn't plugged into a UPS, it won't matter if your ONT has power or not, you're still not going to have any service.  If this power supply is not plugged in, then you will also not have Internet service (unless there's a charged battery).

The cable going down to the ground from your ONT is likely your incoming fiber.