ATT will require you to use their router. They terminate the fiber at an ONT, inside the the house as it requires power. From there it is an ethernet connection to their router WAN port at least the with equipment provided 1-2 years ago. For service install, they will locate the router near the ONT and get the connection up. After that you can move the router anywhere you have an ethernet connection that goes from the ONT to a location. For example, i have an ONT on the wall, ethernet cable to a MOCA2.5 modem to connect over coax to my wiring closet in the house. Ethernet out from the wiring closet moca 2.5 modem into the ATT router WAN. From there i connect to my router with an ethernet cable from the lan port of the ATT router to my router WAN port.
They will normally do an aerial drop of the fiber to the house. If the community requires underground, they should comply. Otherwise, you can provide an underground conduit for them to pull the fiber through to the house exterior wall. Direct burial not recommended unless you protect the cable with an overlay of tile or similar.
i set up an empty electrical box in my wall and drilled through the outside sheathing to provide an entry point. With existing cable, you may be able to use that entry point for the fiber.
it probably will take an "advanced" tech to do anything but a std. drop. You have to talk with the first installer tech that comes out and explain exactly what you want. They have several std installs and the ability to customize but it may take discussion with their supervisor.
Your router will be fine. i have one using ATT fiber at my mother's house. Double NAT doesn't really matter.