Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-spaces have a particular product decomposition. Assuming the spaces in the operad are fibrant we can apply  the coherent nerve construction to obtain an $\infty$-operad with 'one-object' (by this I mean the $\infty$-category associated to the operad has a single equivalence class).  This is Proposition 2.1.1.27 of Lurie's Higher Algebra. 

Q1: Does every $\infty$-operad with 'one-object' come from this construction (up to equivalence)?

EDIT: To clarify, I mean an operad in the classical sense of May (a singly colored simplicial operad, for the multicolored analogue of this question Urs' response says the answer would be yes). 

Let $\mathcal{C}_\mathcal{O}$ be the category of operators associated to an operad $\mathcal{O}$. We know the unit of the adjunction $\mathfrak{C}N\mathcal{C}_\mathcal{O}\rightarrow \mathcal{C}_\mathcal{O}$ is a weak equivalence of simplicial categories. 

Q2: Is $\mathfrak{C}N\mathcal{C}_\mathcal{O}$ the category of operators associated to another simplicial operad $\mathcal{O}^\prime$?