Improve Docker image builds

Some improvements to how we build the Docker image (added in !250 (merged)) can be done:

• Build the Docker image in MRs using a different tag name (e.g., ${CI_REGISTRY_IMAGE}:mr-${CI_MERGE_REQUEST_IID}).
• Add a job to test the built image before pushing to the registry.
• Use resource groups to avoid different jobs clobbering each other's image.