You wouldn’t run 10Gb to a camera. You’d run 1gb PoE+ edge connection from a 10Gb backbone switch to the camera or if your just building a one-off simple network in order to get the camera to a studio then you’d run a 1Gb switch with 1Gb to the camera and 1Gb to the studio.
The key is that the switch connected to the camera and delivering the power needs to be a PoE+ capable switch to deliver the power required by a typical PTZ camera (unless someone is installing an external PTZ camera with night vision as that requires more power than a normal PoE+ switch can deliver - I know that’s not you’re case I’ve just put this here for reference).
I’ve included a link to a UK site which makes it easy to compare switches. The link is for PoE+ switches.
As for routing the cable I’d suggest stick-on trunking up and over. It’s neat, it’s light (especially for the one cable you need), it’s removable, and you can make it more resilient if you need to by screwing through it.
Hope this helps.