Assuming the Mac Mini is compatible (the flat, all-aluminium ones are) the simplest way to move the whole server is do exactly that - copy the entire drive.
Get Carbon Copy Cloner, reboot the Mini in Target Disk mode and connect it to the MacBook with a suitable cable. The Mini will appear as if it was an external drive.
Open Disk Utility and reformat the drive in the Mini.
Open Carbon Copy Cloner, select the MacBook's drive as source, the Mini's drive as destination, click clone and wait a bit. Reboot the Mini and it will be the same as your MacBook.
Yes, you can do this while the MacBook is running. Yes, you can copy the OS between machines. No, you don't need to install OS X before copying.
This used to be a 10-minute job - just pull the drive and install in the new machine. Modern construction makes that nearly impossible now.
You will have to adjust minor things like the Mini's IP address, but it doesn't take long to run through System Preferences and Server.