Automated testing and deployment

Should set up GitHub Actions to:
- Test building website
- Typecheck Nest Bot, Nest CLI, and Cabinet
- Automatically deploy website, Nest Bot, Nest CLI, and Cabinet
- Deployment should be done by SSHing through a dedicated `actions` user on the Nest VM that has permissions to run a script to pull in changes and deploy

Start by locating the website, Nest Bot, Nest CLI, and Cabinet build and typecheck entry points, then determine how deployment through the Nest VM should be scripted. Done means GitHub Actions tests the website build and all three TypeScript projects, and deploys each through the dedicated actions user.

Written by the indexing model from the issue text.