Does $\det(A + B) = \det(A) + \det(B)$ hold?

Well considering two $n \times n$ matrices does the following hold true: $$\det(A+B) = \det(A) + \det(B)$$ Can there be said anything about $\det(A+B)$?

If $A/B$ are symmetric (or maybe even of the form $\lambda I$) - can then things be said?

This does not hold true in general. For even $n$, let $A=-B$ and $\det(A) > 0$, so $\det(A+B)=0 < \det(A)+\det(B)$. Now, consider $A=B$. We have $\det(A+B)=\det(2A)=2^n \det(A) > 2 \det(A)=\det(A)+\det(B)$ for $n>1$ and $\det(A)>0$. Thus, either inequality can hold.

In general, you can't expect a formula for $\det(A+B)$. But sometimes, when you're lucky, you can use the Matrix Determinant Lemma, which says the following:

$$\det(A+uv^T)=(1+v^TA^{-1}u)\det(A),$$

where $A$ is an invertible matrix and $v^TA^{-1}u$ is interpreted as a scalar. Therefore, if $A$ is invertible, and you can write $B$ as $uv^T$ for two vectors $u,v$, then now you have a formula. One could also note that $\det(uv^T)$ is always 0.

If you're looking for a matrix operation which is well-behaved with respect to matrix addition, look for the trace.