So I assume, trains assumes I have nvidia-docker installed on the agent machine?

docker + nvidia-docker-runtime are assumed to be installed
nvidia/cuda docaker image is pulled when requested (like any other container image)

Moreover, since I'm going to use Task.execute_remotely (and not through the UI) is there any code way to specify the docker image to be used?

Sure, task.set_base_docker(docker_cmd='nvidia/cuda -v /mnt:/tmp')
Notice that you can not only pass the docker image but also provide the docker with execution parameters like volume mounts or environment variables, etc.