It depends if you are trying to have convience or save money. Digital Ocean does charge for snapshots, even if the price is small: https://docs.digitalocean.com/products/snapshots/. If tghe site is small, I’d also take a backup just incase.
Assuming your mailgun access did not change, that should not be required. As for DNS, if the hostname has not changed, when you spin it back up, it should also be fine. I would do an backup/import if only because that way you can have the setup process run again
since you want to save costs, you can better create a backup. You can store it on your own computer / Dropbox / whatever and not be billed. However, please note #2b below.
2a) A new droplet implies a new IP address so you do need to update DNS.
2b) Restoring a backup restores everything EXCEPT for mailgun settings (and a few other things like your SSL certificate). You do need to configure Mailgun using your app.yml or the setup script. I think it is the most convenient to save the original app.yml next to the backup.