I recently upgraded my home network to 10Gbps. As it was when it was only 1Gbps, everything that could be using a wired connection is using one. The WAPs (now R770s) are used to provide Internet for mobile devices, and a few other IoT devices that don't have a physical Ethernet port.

WAPs are independent of routers, but many low-end consumer routers provide integrated WiFi, which can usually be turned off if desired.

Most WAPs provide a means to set the TX power levels, and some newer ones will automatically negotiate with the client(s) to use the minimum required power to close the link (just as cellular phones do).