Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this?
I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal length) cannot be recovered from the (extended) dynkin diagram of $su(5)$. So, if it is a subalgebra, the cartan generators of $sp(4)$ is niet a subset of those of $su(5)$...
No, for dimension reasons. $\dim \mathfrak{sp}(4) = 4 \cdot 9 = 36$ but $\dim \mathfrak{su}(5) = 5^2 - 1 = 24$. The smallest $n$ such that $\mathfrak{sp}(4)$ could embed into $\mathfrak{su}(n)$ on the basis of dimension alone is $n = 7$.