We have a MS SQL Server installed on a stand alone server.
We would now like to add the server into a domain.
Is this possible? Do we have to do anything with the rights to the database after the move?
"Do we have to do anything with the rights to the database after the move?"
If you want to give domain users rights to the different DBs on the server, you'll have to add them in the "logins" and give them roles. That won't happen by its self.
I would also double check your effective GPO policies on your domain, and make sure when you add the server to the domain, and it takes the GPO update, it's not going to effect things like Windows Firewall and lock/open ports that you weren't intending on.