Are there any bike GPS computers / devices that can create routes themselves out on the road? In other words, I would want to be able to be in an arbitrary location and tell the device that I want to go to a specific destination (by entering an address or town name) and it would calculate a route from where I am to that destination and ideally re-calculate if I went off-course.
From researching such devices, it seems that you need to prepare the routes at home and then physically copy them to the device, or download it through an app onto the device. I could cope with having to use a smart phone to create the route (by specifying a destination) and transfer that the GPS device, but I want to be able to do this out on the road.
I do not have any experience at all with any bike GPS devices other than just purely to tell me speed (presumably all bike GPS devices can display the current speed). I only have experience with car sat-nav systems and the above is how they work.
Yes.

The Garmin Edge 530/830/1030 can be loaded up with extensive map data, and can give you live routing, even without a cellular connection.
Wahoo's ELEMNT ROAM can also do routing on the fly.
The Sigma Rox 12 also has extensive navigation functions.
Probably others.

There are some smartphone apps that can do this—I think the best-known are Ride with GPS and Komoot. The idea with these apps is mostly that you would use them instead of a dedicated bike computer, not together with one, but you can do that too. They create a route stored in the cloud that you can sync down to your bike computer (using the phone as a hotspot, of course).