So my cousin wants to build his case inside an NZXT H230 case but his Mobo doesnt have USB 3.0. Does the cases USB 3.0 Plug have a USB 2.0 adapter connected to it already to run at 2.0 speeds? Havent been able to find the answer anywhere.
it is possible to take (as per your image) a USB 3.0 pinout on the motherboard and connect it to USB 2.0 pins on the USB port. That is what downward compatibility is all about. The newer USB 3.0 has all the pins (and more) of USB 2.0 - that is why you can plug a USB 2.0 device into a USB 3.0 port and it will work just fine.
What you cannot do is manufacture or pretend to have the USB 3.0 additional pins if you start with USB 2.0.