@user44191: If $\alpha>0$ and $\delta<0$, then $\lambda\mapsto \lambda\alpha+\frac{\delta}{\lambda}$ is a continuous function tending to $\infty$ for $\lambda\rightarrow\infty$ and to $-\infty$ for $\lambda\rightarrow 0$. So for any given $\beta, \gamma$, there is some $\lambda$, such that $\lambda\alpha+\beta+\gamma+\frac{\delta}{\lambda}=1$. So for any such tuple there is some function $F$ (namely $\lambda\mathrm{id}$ for a suitable $\lambda$), for which this particular $H$ satisfies the functional equation.

For any given map $h$ the probability can be computed by a weighted coupon colector problem. The distribution of $h$ can also be computed. The combination of both counting problems can become quite difficult, depending on what parameter range and what accuracy you look at. If $\delta$ is much bigger than the probability that $h$ is not surjective, you can neglect all "strange" partitions occurring as pre-image of $h$, and some standard coupon collector results suffice. If $\delta$ is close to that probability, the problem will become pretty difficult.

@user44191: The random customers could either all be the one customer, that likes card 1, or all of them could be one of the two customers, who like card 2.

Take a small rectangle around the real line, and use Rouche to count the roots of the function within this rectangle. Use sing changes to locate real roots. If you make the rectangles sufficiently small, you will find as many real roots in the interval as there are complex roots in a neighbourhood of the interval, and you are done. The problem is that a real double root, two real roots close to each other and two complex conjugate roots close to each other cannot be distinguished. If all roots are simple, the algorithm will terminate one way or the other, although you cannot predict when.

I don't think this is related. Standard numerics tells you how many complex roots there are in an open domain, and you can detect sign changes on the real line. So the other question is equivalent to the problem whether we can distinguish a multiple root from a simple one. As $t$ in the other question is arbitrary real, one would need some result such as $\mathrm{tr}\deg\mathbb{Q}(t_1, \ldots, t_n, e^{t_1}, \ldots, e^{t_n})\geq n$, which is as far as I know not known.