Yeah, you can host multiple domains/websites, there's no restriction on that. But you will have to do the setup and maintenance. Digital Ocean provides the base server and then you can put what you like on top of it (although there are some templates that provide various software stacks including LAMP on Ubuntu for example).
To manage it more easily you may want to install a control panel (e.g. cPanel, zPanel etc.) but you'll still need to have some understanding of how to manage the server. You'll also have to manage the FTP server and user access yourself, DO won't provide any support for that.
As for MX records, if you host your customer's domains using DO you'll be able to change them as you wish. Although you won't be able to give the customers access to change the records (unless you build your own control panel on top of DO's API.)
Do say if you have any more questions!