Why not? I did. Many others have also pointed email elsewhere. It’s true there are no good Free options. The problem with free is that it attracts every spammer on the planet.
With Office 365 Home the move was simple. Create your account online and go through the setup wizard, when you get to DNS it looks up the domain you want to move and tells you exactly what to add, and if conflicting records exist. It’s really pretty straightforward, you make the changes it directs you to make and then wait for propagation, once you can return to that page and see all green checks your email is moved away from dreamhost.
and yes you can run multiple domains and have unlimited aliases and shared mailboxes that don’t cost extra.
you can also install your own copy of roundcube and customize it however you want. I doubt the stock shared dreamhost install of roundcube has that installed, it’s likely a plugin.