You can indeed get video, control, and power all through the Ethernet connection on the SDI/NDI and ZCAM models of our cameras. The only cameras we have that do not support PoE are the webcam and the USB models.
Both OBS and vMix would be able to access either the RTSP or NDI video over the network. The PC running the software would just need to be connected to the same network as the camera. So you would not need to use the SDI or HDMI outputs from the camera to bring in the video.
There are multiple options for controlling the cameras over the network.
There is a mobile camera control app.
vMix also has built in PTZ control if you get the 4k version or better.
For a hardware option, we do have an IP based joystick controller, the PT-JOY-G3. It would just need to be connected to your network via Ethernet like the camera. The IP joystick can also be powered through PoE.
Please let us know if you have any other questions.