To follow on what @MichaelOjeda said, once you have an API Token, the curl request to add a domain via the API is:

curl -X POST -H "Content-Type: application/json" -H "Authorization: Bearer APITOKEN" -d '{"name":"example.com","ip_address":"1.2.3.4"}' "https://api.digitalocean.com/v2/domains"

For NEW droplets, you can automate this by pasting a user-data cloud-config script in the User Data field on the Droplet Create page:

#cloud-config
runcmd:
  - export DROPLET_NAME=$(curl -s http://169.254.169.254/metadata/v1/hostname)
  - export DROPLET_IP=$(curl -s http://169.254.169.254/metadata/v1/interfaces/public/0/ipv4/address)
  - 'curl -X POST -H ''Content-Type: application/json'' -d ''{"name":"''"$DROPLET_NAME"''.example.com", "ip_address":"''"$DROPLET_IP"''"}'' -H "Authorization: Bearer API_TOKEN" https://api.digitalocean.com/v2/domains'

replace example.com and API_TOKEN

The #cloud-config line specifies that you are using cloud-init language (which interprets this as YAML), runcmd: specifies that the following lines should be run in shell when server first starts.

The two export lines set the droplet’s IP address and name as environment variables, and the final line makes the HTTP request to create a new domain via the API.

(All the extra quotation marks are due to making YAML and JSON play nice together)

This is possible, but you would need to do it through the API. There isn’t (and probably will never be) a option to do this in the portal. Here’s the link to the documentation: https://developers.digitalocean.com/documentation/v2/#domain-records