Given a locally compact Hausdorff space $\Omega$.
Consider the C*-algebra: $$\mathcal{C}_\infty(\Omega):=\left\{\varphi\in\mathcal{C}(\Omega):\|\varphi\|_{K^\complement}\stackrel{K\to\infty}{\to}\infty\right\}$$
Generated C^*-Algebra: $$\mathcal{A}(\chi):=\overline{\bigg\{p(\chi,\chi^*):p\in\mathbb{C}[X,X^*]\bigg\}}$$
It admits cyclic elements: $$\chi\in\mathcal{C}_\infty(\Omega):\quad\mathcal{A}(\chi)=\mathcal{C}_\infty(\Omega)$$
I don't think so, or? (Example?)
Not in general. If there exists $\chi\in C_\infty(\Omega)$ such that $\chi$ is injective then Stone-Weierstrass implies that $\mathcal A(\chi)=C_\infty(\Omega)$.
But for "most" $\Omega$ there is no such $\chi$, and if $\chi$ is not injective it's clear that $\chi$ does not generate $C_\infty(\Omega)$: If $\chi(a)=\chi(b)$ then $f(a)=f(b)$ for all $f\in\mathcal A(\chi)$, hence $\mathcal A(\chi)\ne C_\infty(\Omega)$.