Existence of a common minimizer Do 
\begin{equation}
\sum_{i=1}^{3}\left|\lambda_{i}\left(\operatorname{diag}\left(1,3,5\right)-U\operatorname{diag}(2,4,6)\,U^{T}\right)\right|
\end{equation}
and
\begin{equation}
\sum_{i=1}^{3}\sqrt{\left|\lambda_{i}\left(\operatorname{diag}\left(1,3,5\right)-U\,\operatorname{diag}(2,4,6)\,U^{T}\right)\right|},
\end{equation} where $\lambda_{i}(\cdot)$ is the $i$-th eigenvalue of a symmetric matrix, as functions of orthogonal matrices $U$ acheive their minimums at
the same orthogonal matrix?

Let $D=\operatorname{diag}(1,3,5)$. Then the first minimisation problem can be rephrased as
$$
\min\limits_{U\in O(3)}\sum_{i=1}^{3}\left|\lambda_{i}\left(D-UDU^T-I\right)\right|
=\min\limits_{U\in O(3)}\sum_{i=1}^{3}\left|\lambda_{i}\left(D-UDU^T\right)-1\right|.\tag{1}
$$
Note that $D-UDU^T$ has real eigenvalues (because it is real symmetric) and its trace is zero. Hence $\lambda_1+\lambda_2+\lambda_3=0$. Now it can be shown that for any two real numbers $a$ and $b$, $|a-1|+|b-1|+|-a-b-1|\ge3$ and equality holds if and only if $(a,b)$ lies inside the triangle $R$ bounded by $a,b\in[-2,1]$ and $a+b\ge-1$. Therefore, if there exists a real orthogonal matrix $U$ such that the eigenvalues of $D-UDU^T$ are real and some two of them lie inside $R$, then such $U$ is a global minimiser to $(1)$.
For your second minimisation problem, it can also be shown that for any two real numbers $a$ and $b$, $\sqrt{|a-1|}+\sqrt{|b-1|}+\sqrt{|-(a+b)-1|}$ is minimised when $(a,b)=(1,-2),(-2,1)$ or $(1,1)$. So, if we can find a real orthogonal matrix $U$ such that its eigenvalues are equal to $1,1,-2$, then $U$ is a global minimiser with at which the objective function value is $\sqrt{3}$. Since such $U$ also satisfies the aforementioned sufficient condition in terms of $R$, it (if exists) is a common global minimiser to both minimisation problems.
So it boils down to finding a real orthogonal matrix $U$ such that the spectrum of $D-UDU^T$ is $(1,1,-2)$. Or equivalently, we want to find two real orthogonal matrices $P$ and $Q$ such that $PDP^T-QDQ^T=\operatorname{diag}(1,1,-2)$. Apparently this is solvable and the numerical value of the minimiser can be obtained by applying the conjugate gradient method on a manifold, or by the conjugate gradient method with successive linearisations. Yet an approximate solution can be found by computer simulation. After running nearly 500,000 simulation trials, I found that the common minimiser is given by
$$
U=\pmatrix{
\sqrt{\tfrac{45}{64}} &-\sqrt{\tfrac{5}{32}} &\tfrac38\\
\sqrt{\tfrac{9}{32}}  &\tfrac34              &-\sqrt{\tfrac{5}{32}}\\
-\tfrac18             &\sqrt{\tfrac{9}{32}}  &\sqrt{\tfrac{45}{64}}
}.
$$
