Let me answer Question 2. The answer is in negative: there exists an upper bound for majority coloring numbers of all tournaments. I will not care about the sharpness of the bound.

Let $G$ be a tournament on $n$ vertices, and let $d_1\leq d_2\leq \dots\leq d_n$ be the in-degrees of its vertices (denote the vertices $v_1,\dots,v_n$ respectively). We want to find its majority coloring in $3k$ colors.

Firstly, notice that the first $m$ vertices induce a sub-tournament with $m\choose 2$ edges. This means that $d_1+\dots+d_m\geq {m\choose 2}$, which yields $d_m\geq \frac{m-1}2$. Now we paint $v_1,\dots,v_{2k}$ with the last $2k$ colors; all the remaining vertices will be painted with the first $k$ colors, so the conditions for $v_1,\dots,v_{2k}$ are satisfied automatically.

Each of the remaining vertices has in-degree at least $k$. Consider a random coloring of them in $k$ colors. Now, for each $i>2k$, the probability $p_i$ that at least half of $\mathop{\rm In}(v_i)$ are painted with the color of $v_i$ can be bounded by Hoeffding's inequality as
$$
p_i\leq \exp\left(-2\left(\frac12-\frac1k\right)^2 d_i\right)^2
\leq \exp\left(-\frac{(k-2)^2}{2k^2}(i-1)\right)
$$
(the estimate may become even better if $\mathop{\rm In}(v_i)$ contains some of the first $2k$ vertices).
So the probability that at least one vertex violates the required condition does not exceed
$$
\sum_{i=2k+1}^n p_i
\leq \sum_{i=2k+1}^n \exp\left(-\frac{(k-2)^2}{4k^2}(i-1)\right)
<\sum_{j=2k}^\infty \exp\left(-\frac{(k-2)^2}{4k^2}j\right)\\
=\exp\left(-\frac{k(k-2)^2}{4k^2}\right)\cdot
\left(1-\exp\left(-\frac{(k-2)^2}{4k^2}\right)\right)^{-1}.
$$
This last bound tends to $0$ as $k\to\infty$, so it is less than $1$ for some value of $k$. This value fits.

**Remarks.** 1. One may try to improve this estimate by making something different with the vertices of low in-degree, e.g., introducing them into a general scheme and computing the sharp values of $p_i$ for these vertices. However, I doubt that a sharp bound may be obtained in this way.

- The estimates from the beginning do not work for arbitrary digraphs. However, we may see some conditions a hypothetical example should satisfy.

Firstly, a minimal example should not contain vertices of low total degree, since one may color the graph without such vertex, and then paint it with a color different from all its neighbors' colors. On the other hand, this graph should contain *many* vertices of *relatively small* in-degree, since otherwise the estimate using Hoeffding's inequality would work.