To stream with a camera such as the a7III via HDMI, you'll need another piece of hardware. An example of this kind of thing would be "elgato" which produce different devices for this exact thing. The simplest way is to buy a USB dongle which has a female HDMI and a male USB. Plug it into your pc, plug the camera into the dongle and boom. The camera will be interpreted as a Webcam and can thus be used in OBS, Streamlabs etc.
To prevent the camera from turning off: There's different answers I could find. Some say leaving the battery/cassette-slot open will prevent the cam from going to standby. Others say you need to disable "remote control" in the settings. Others again say it's not possible. My guess is that there's either a setting, a way to trick the hardware or a software-hack that can help you out.