You get charged the hourly fee for your droplets, up to the maximum of the monthly charge. Your droplet being shutdown still gets charged as all the resources, and IP and all your data is still there.
So for the $10/mo plan you'll actually be paying $0.015/hour, but with a monthly cap of $10/month on that droplet

You're required to pay the fees on the 28th of every month, so if your month is really 30 days, you'll lose 2 days worth though that's nothing really.

Yes, DigitalOcean is a perfect replacement for a VPS from a traditional web host.

The experience may not be as user-friendly, but a simple website can run perfectly fine on a $5 droplet and if you (for whatever reason) need to serve more visitors, your website or a web service monitoring your website, can programmatically expand your ability to serve traffic by way of adding more droplets, or increasing the droplet size to say the $10 or $20 one.

That's the biggest fundamental difference between a cloud virtual machine from DO and a virtual private server from godaddy. The flexability of management through a software API. That, and hourly billing. Because if for example you went on a weekend trip and your site exploded in hits that same weekend, it could have ramped up to the $20 plan for two days, and then dropped back down again all while you were fishing.

Traditional hosts would require manual intervention, a purchase of a whole month of the $20 plan, and the setup wouldn't be as responsive to the surge as a cloud virtual machine can be.

If you like the control panel experience from godaddy, you'll need to buy a license directly from Plesk or CPanel, and there are many many articles hosted at DigitalOcean that help with various server management needs.

Good luck!