Let me address your first query, 'What is runner actually for?'
The project file is stored along with a .gitlab-ci.yml file. Note that, .gitlab-ci.yml defines the stages that the CI/CD pipeline has, it also defines what to do in each stage. It consists of build, test, and deploy stages. Within each stage multiple jobs can be defined.
e.g.: In the build stage, we may have 3 jobs to build on debian, centos and windows (in GitLab glossary build:debian, build:centos, build:windows). A GitLab runner clones the project read the gitlab-ci.yaml file and do what he is instructed to do.
In other words, GitLab runner is a Golang process that executes some instructed tasks.
Now, let us see where this is meant to be installed.
We can install a GitLab runner in the desired environment listed here, https://docs.gitlab.com/runner/install/ or we also can use a shared GitLab runner that is already installed on GitLab's infrastructure.
Let us discuss if the directory it is run in actually matters?
Yes it does matter. Every task executed by the runner is relative to CI_PROJECT_DIR defined in https://gitlab.com/help/ci/variables/README.
Now let us see where it executes it's script commands. At root? Do we need to set our executor to docker? Should we just set it to shell, pull the image, and build it?
A runner can have multiple executors such as docker, shell, virtualbox etc, docker is the most common one among all. If you use docker as the executor you can pull any image from docker hub or your configured registry and you can do loads of stuff with docker images. In a docker environment normally you run them as the root user. Refer to this official doc for more https://docs.gitlab.com/runner/executors/README.html