some sort of dashboard with monitoring and statuses of all my services

See if Uptime Kuma suits your needs.

Are there any major no-no things a newbie might do with Docker/selfhosting that I should avoid?

Allow password based SSH authentication, you should look into key based authentication

I wanted to have it behind a domain, how would I achieve it?

Use a reverse proxy (like caddy) which serves different content based on domain name.