What I need to know is if it were possible to give the outside company the information for only the new database, such as user/pw and database name, and can they access and work on the database without the need of going through cPanel?
Yes, you can. You need to give them the necessary credentials (server info, IP or host name, username, pass, database name). They need to give you the IP(s) that they'll be connecting from. Then you go to "Remote Mysql" in cPanel and add them there.
Now they will be able to connect to your mysql server and do anything they need to.