Kuhn Tucker Conditions with fewer non-negativity constraints than number of variables
Non-negativity constraints have nothing special and are not critical for the general validity of the Karush-Kuhn-Tucker approach. First, realize that we could have $x \geq a &gt;0$ and then we could write $x-a \geq 0$ and view this "non-negativity" constraint as just one more inequality constraint on the solution.

When a decision variable is weakly bounded, the new phenomenon that emerges is that we may have the partial derivative non-zero (strictly negative, for the maximization case), if at the solution this decision variable lies on the boundary.

If it is unbounded, than we must have the first-order derivative equal to zero. As another question pointed out, this may create issues in certain contexts, but it does not invalidate the approach.

As for your second question, indeed strict inequalities may create problems, but don't jump immediately to a "not well-defined" heavy statement. The problem will arise only if (stars denote the optimal solution)

$$\frac {\partial F(s^*,x^*,y^*,z^*)}{\partial s^*} &lt;0$$

which would "push" the solution vector towards the unreachable zero-lower boundary for $s$. To avoid that, we require that at the solution the first-order derivative w.r.t to $s$ equals zero, and see whether we can indeed obtain a solution vector. If we can, it means that the $s&gt;0$ constraint does not even "tend to" be binding, so you can ignore it.