He says no paperwork is required with exception of latest statement from the mortgage company.

The LLC has been operational for over a year, filed taxes, has all the accepted paperwork from the state including a certificate of good standing, but they didn't like nomenclature. 

If it is an option to bonus landlords that would definitely ease some of the blow of losing a month rent waiting on their paperwork.

It was about $159 for them to do all the paperwork and submit it to your state.

Definitely have a lawyer draw up the paperwork to ensure all your bases are covered. 

Why not transfer what he's paid to the new property, pay extra (assuming the second house is worth more) and a fee for transferring as it will cost money to do all the paperwork and everything else.

Again, it depends on what you are doing to determine if the extra paperwork, registrations, bank accounts, agreemtns, etc. is worth it -- unless you just want to do it. 

Make sure you get the inspection and ideally be there for it so you can ask questions.As for the 4 weeks, it takes time for paperwork including title searches, inspections and the person living there to find a place to move to.

I guess I'm looking for the legal stuff/paperwork stuff that needs to happen once you start renting out a unit or what needs to happen before you can rent out a unit. 

I would contact the attorney that helped you come up with the paperwork.