Modelling of minimal NOR-circuit problem with CP
This is a logic synthesis problem.  Specifically, without the "fan-out of 1" limitation, this looks like the circuit minimization problem.  Circuit minimization is $\Sigma_2^P$-complete, so you should not expect any efficient algorithm for it.  There are many algorithms in the literature that you can explore, including Karnaugh Maps, Quine-McCluskey, Espresso, and others.  I wouldn't consider constraint programming the obvious first way to approach this problem.

If you have a requirement to use constraint programming for some homework exercise, well, you should do  your own exercise, but here is a hint: you should be able to tell whether there exists a circuit of depth $d$ and width $w$ by building a $d \times w$ grid of NOR gates, then using decision variables to represent which edges are used to connect these gates.  How could you use that to solve the original problem?

Limiting fan-out to 1 seems unrealistic, as gnasher729 explains.