Stone-Čech Compactification of The integers is $T_5$? Since $\beta\mathbb{N}$ is $T_2$ and compact we can conclude that it is $T_4$. However, what about $T_5$? My only guess will be that $\beta\mathbb{N}-\mathbb{N}$ is not normal. Is this true?

No, as all points of $\mathbb{N}$ are isolated ,it's an open subset of $\beta\mathbb{N}$ (in general any locally compact Hausdorff space $X$ will be open in $\beta X$). So its remainder $\beta\mathbb{N} \setminus \mathbb{N}$ is a closed subset of $\beta \mathbb{N}$ hence compact + $T_2$ thus normal.
this paper  has a construction of non-normal subspaces. 
