About a universal (functionally complete) function producing a constant
The claim is false if you're allowed to use constants. Take $f(x,y,0,0) = f(x,y,1,1) = 1$ and $f(x,y,0,1) = f(x,y,1,0) = x \text{ NAND } y$. If you're not allowed to use constants, then every function that you can get by composing $f$ with itself will have the property that if all inputs are 1, then so is the output. NAND doesn't have that property.