Yes, you can add your domain to DigitalOcean and manage your DNS records here, however, you can’t transfer the domain over in terms of DigitalOcean being a registrar. Having said that, managing your domain’s DNS here is perfectly possible and viable.
To do so, you’ll need to change the Nameservers of your domain over at your current registrar to :
Then you should be able to add it to your control panel. There is really good documentation following all the steps to do so which you can check out here :
As for your other question, yes, you can store your website on your DigitalOcean droplet. You’ll need to configure it on your own, in terms of installing Apache, MySQL, PHP however yet again, DigitalOcean has really good tutorials that can help you in that direction. In your case, I feel the best tutorial will be the following :
A "LAMP" stack is a group of open source software that is typically installed together to enable a server to host dynamic websites and web apps. This stack typically consists of the Linux operating system, the Apache web server, a MySQL database, and PHP, a dynamic content processor. This tutorial serves as a guide for installing these components on an Ubuntu 18.04 server.