You can do both. The important thing is to register with one of the performance rights agencies (ascap/bmi/sesac) as a publisher as well.
So you can have Sleepy Productions and Sleepy Publishing.
If you sign someone to your label, you don't automatically own their publishing, unless it's spelled out in the contract that they sign.
You should really read the book "Everything You Need To Know About The Music Business" by Donald Passman. That will answer all of the questions you could ever have.
Publishing is very confusing and very tricky. Then you're throwing business on top of that, with labels, signings, etc. Gotta know what you're doing before you go dealing with all of that stuff.
Good luck.