The first question was already answered in the comments, take $X$ to be the Cantor set.
If a topological space $X$ has a dense discrete subspace, what properties must it satisfy? For example, is $X$ totally disconnected, or does this only apply for compact Hausdorff spaces? What about local compact Hausdorff spaces?
Let $Z$ be any non-empty separable compact Hausdorff space, take a continuous surjection $f:\beta\mathbb{N}\setminus\mathbb{N}\to Z$ (this is possible since $\beta\mathbb{N}\setminus\mathbb{N}$ contains a copy of $\beta\mathbb{N}$ and $Z$ is separable), then there exists a compactification $X$ of $\mathbb{N}$ induced by $f$ such that $X\setminus \mathbb{N}\cong Z$ (since $\mathbb{N}$ is locally compact; theorem 3.5.13 in Engelking). So a compactification of $\mathbb{N}$ can contain any separable Tychonoff space you want.
In fact if $Z$ is any non-empty compact Hausdorff space, take an infinite discrete space $Y$ and a map $Y\to Z$ with dense image. Then since $\beta Y\setminus Y$ contains a copy of $\beta Y$, there is a continuous surjection $f:\beta Y\setminus Y\to Z$ and so a compactification $X$ of $Y$ such that $X\setminus Y\cong Z$. So a compactification of discrete space can contain any Tychonoff space you want, and any non-empty compact Hausdorff space arises as remainder of such compactification (here compactness is essential since a discrete space is locally compact).
So as you see, it doesn't have to be totally disconnected even if compact (e.g. $Z = [0, 1]$), and there is not much hope for its "properties". It can basically be anything.
