The reason for installing CB first is because the DT Register install will create the additional CB profile fields for you. By default, CB does not have address, city, state, etc. You can either manually create the profile fields in CB, or just reinstall DT Register. If you uninstall and reinstall DT Reg (after installing CB), you will not lose any data. No worries there. Only the files are replaced. So if you have made any changes to language, CSS, etc, just backup those files so you can reload them after the reinstall.
The integration works just by enabling the CB integration parameter in the main Configuration of DT Reg. There is an optional CB Plugin that came with your DT Reg package that you can use if you want. It will add a tab to the user profile to show the registration history of that user... what they registered for and when.
For the CB name style, you need to use First Name / Last Name... NOT Single Name. Hope that helps!
Yes, DT Reg will create the additional profile fields in CB. You only need to reinstall the component. It doesn't matter when you install the CB Plugin. It is found in the DT Reg package. The included readme file explains what every item in the package is and what to do with it.