AFAIK there is no way to turn off the OLED display. I haven’t found the OLED display bright, but I am not sensitive to it (it has about 5% white pixels on a black background). Turning off the OLED would maybe be a cool feature request.
The LEDs are bright, and before the possibility to turn them off I have used black electrician tape to cover them. They will still leak some light through the vents. The normal way to turn the LEDs off is in the AirGradient Cloud API. That is something manual which you will normally do once.
Home automation systems like OpenHAB (https://www.openhab.org/) and Home Assistant (https://www.home-assistant.io/) let you automate more. I have written the AirGradient plugin for OpenHAB, and it lets you set up rules to e.g. turn off the LEDs at certain times or when you turn off the lights in the room. I have no doubt Home Assistant will get similar functionality also (it is currently being written).
The difference between OpenHAB and Home Assistant (according to me, and I am biased) is that Home Assistant has a bigger and more vibrant community, while OpenHAB seems to be built on a more robust foundation. I am hoping that will give me less work running the system over time
Those home automation systems are something you normally install on a RaspberryPi to connect many different “smart home” devices in your home.
Currently this functionality needs special firmware on the devices to work when you communicate with it directly (without going through the cloud), but the work to fix this has been started, and is needed for both home automation systems, so I expect that to be resolved in some weeks.
Flashing official firmware is pretty simple (connect the device to your computer, press some buttons in a browser). Setting up a home automation system is harder, and something I would only recommend for enthusiasts