Bayes theorem for mutations
I don't understand your calculations and I don't understand why you're trying to use Bayes formula. I don't know the $\frac{1-p}{2-p}$ formula and I don't understand what it is supposed to calculate. It seems to me that you're overthinking a simple problem.

We don't have all the information and need to make a bunch of assumptions but if I understand the question correctly then..

Probability that the father carries the mutant allele

I neglect the probability that a new mutation occurred and that the son is the first carrier of the mutant in the lineage. I assume that we have a probability of 0 of being sick if we don't carry the disease. Because MEN is a pretty bad disease, the autosomal dominant alelle causing the disease is probably kept at pretty low frequency. From wiki, I read that MEN 2A has a frequency of 1/40000. This means that the probability of being homozygous mutant is $\left(\frac{1}{40000}\right)^2 ≈ 10^{-9}$. I think we can safely assume that the sick kid is heterozygous. Given that the kid is heterozygous, the mutant allele comes either from the mother or from the father with equal probability. So the probability it comes from the father is $0.5$ and the odds are $\frac{p}{1-p}=\frac{0.5}{1-0.5}=\frac{0.5}{0.5}=1$

Knowing that the father is heterozygous, what is the probability that a given kids receives this mutant

By the law of segregation, the probability is 0.5 and the odds are 1.