I think DigitalOcean probably bills Dply for one hour minimum so if you could immediately destroy and recreate then you'd quickly add up to a lot more OpEx.

DigitalOcean does it's billing based on a per-minute basis based on their hourly rates. We have already implemented a feature in our backend code to allow users to "expire" free servers and are working on adding that to the dashboard UI and this option should be available this weekend.

That's not what their website claims.

How am I billed?

Each Droplet is billed per hour up to its monthly cap.

Do I have to pay the cost of the server every time I create a new one?

No—you only need to pay for 1 hour of usage. For example, spinning up a new 512MB server just to test something for a couple of minutes will only cost you $0.0074

https://www.digitalocean.com/pricing/#business

There seems to be a "Rebuild" button that allows you to recreate the server from another base image.

Yeah, I didn't notice that you could change the OS when rebuilding a server. It should also let you edit the cloud-init script thing.