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 and a weak equivalence of such operads (a singly colored simplicial operad, for the multicolored analogue of this question Urs' response says the answer would be yes). So one side I'm allowing arbitrary weak equivalences of $\infty$-operads and on the other I'm only considering weak equivalences between simplicial singly-colored operads.

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 $\mathcal{O}^\prime$?