Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (*see comment by José*).

My question is, can we determine, by either adding or multiplying digit-wise (mod 10) the digits of $\pi$ and $e$, whether the resulting numbers are transcendental?

Suppose $p_n$ and $e_n$ are the $n$th digits of $\pi$ and $e$ respectively. Let $d_n = p_n + e_n \ (\mbox{mod } 10)$. The new number $t$ will have $d_n$ on the $n$th digit. We can similarly define $t$ with $d_n = p_n \cdot e_n \ (\mbox{mod } 10)$. Is $t$ transcendental?

More general version of the question:

Since it might be very difficult to determine whether two individual numbers are transcendental, here is a more general question:

If we add or multiply the digits of $\pi$ or $e$ digit-wise (mod $n$), where $n$ is a digit from 2-10, which of the resulting numbers are transcendental?

In the general case, define $d_n = p_n + e_n \ (\mbox{mod } k)$ where $k$ is an integer in $\{2,3,4,5,6,7,8,9,10\}$. In this case, we can use a choice function to determine $k$, or determine $k$ from $n$ in some suitable way. In the general case there are many possibilities for the different kinds of numbers that can be formed.