First
At DigitalOcean should I choose the app: https://marketplace.digitalocean.com/apps/discourse
Or should I choose a vanilla ubuntu 18.04 and then install it using https://github.com/discourse/discourse/blob/master/docs/INSTALL-cloud.md
Second
The best way to move the content. Just backup and then import? Using
/admin/backups
Using https://github.com/discourse/discourse/blob/master/docs/INSTALL-cloud.md would be the way to go because that is the only officially supported method for now 
Yes, see Create download and restore a backup of your Discourse database for more details.
btw: when moving to digtitalocean I will change domain to a subdomain
studmed.dk —> forum.studmed.dk
anything I should be aware of? Except the dns record 