What is the height of the IKEA BILLY plinth?
Not sure if I would call anything associated with the Billy a "plinth" (in the sense that it is sturdy)... I have several and the structure is really all in the sides (and how vertical you keep them). If you put support anywhere under it that isn't directly supporting the sides, you are gonna have a bad time.

The material (hesitate to call it wood but i guess it is) you see at the front below the first shelf is just a very thin piece of fiberboard. The first (lowest) shelf is held in the same way the middle shelf and top is, via two cam screws and plastic cam locks in the shelf. For what its worth, the space under the first shelf measures 11.75cm high.

As to your plan to use an X brace, that would prevent the unit from swaying but remember this is all fiberboard (and rather loosely pressed at that) so any single point support that exerts a lot of force is going to split it open like a soft burrito.  The most popular way to make the Billy into something sturdy is to gang it together (my installation has 5 of them across the wall, each anchored to the wall at the middle and at the top, and to the one next to it with screws in both directions).  

And lastly, why would you remove the front face and put feet on it with the intention that it will become easier to clean under? IMO the purpose of that little face is to keep dust and other random bits from settling back there in the first place.