Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ does there exists an IRREDUCIBLE family of curves in $\mathbb{P}^3$ such that $C$ is a general fiber and there exists a fiber which is the union of lines (not necessarily reduced)?

I thought somebody had already answered this question for you in an earlier post. Assuming that by "union of lines (not necessarily reduced)" you mean "connected onedimensional scheme whose irreducible components are lines", then then answer is yes. Take any codimension $2$ linear subspace $M$ that is disjoint from $C$, and let $L$ be a line that is disjoint from $M$. Consider a $\mathbb{G}_m$action on $\mathbb{P}^n$ that has $L$ and $M$ as eigenspaces. For the induced action of $\mathbb{G}_m$ on the Hilbert scheme, one of the "limit points" of the orbit of $[C]$ will be a scheme supported on $L$. 

