I would like to ask about how we can talk about Algebra over little $n$-disk operad $D_n$ in a greater generality outside of $\textbf{Top}$.
I know that in the topological context, an $E_n$-operad is a topological operad $P$ which is weakly equivalent to $D_n$ and an $E_n$-algebra is a topological space with a structure of an algebra over such $P$.
But I read the term such as "$E_2$-algebra in $\textbf{Ab}$, $\textbf{Cat}$, etc..." was wondering what it precisely mean by $E_n$-operads and $E_n$-algebras in general symmetric monoidal model category other than $\textbf{Top}$.
I know that with a symmetric monoidal functor $(F,F_2,F_0): \textbf{Top} \to W$ where $W$ is a symmetric monoidal model category, we can transfer a topological operad as an operad over $W$ by the Base Change Theorem. i.e. if $(A, m, i)$ is a topological operad where $m$ and $i$ is the composition map and the unit map respectively, we get an operad $(F(A),\overline m, \overline i)$ over the symmetric monoidal model category $(W, \otimes, I_W)$ by $F(A)(n) = F(A(n)) \in W$ and
$\overline i: I_W \to F(A(1))$ is the composite $$\require{AMScd} \begin{CD} I_W @>\ F_0>> {F(I_V)} @>\ F(i)>> {F(A(1))} \end{CD} $$ and $\overline m: F(A(n)) \otimes F(A(k_1)) \otimes \cdots \otimes F(A(k_n)) \to F(A(k_1+\cdots + k_n))$ is done similarly.
And in the new context of $W$, an $E_n$-operad is an operad $P$ over $W$ weakly equivalent to $F(D_n)$ and an $E_n$-algebra is an object of $W$ with a structure of an algebra over such $P$.
For example, we can obtain a $\textbf{Set}$-operad from the path components functor $\Pi_0: \textbf{Top} \to \textbf{Set}$ or a $\textbf{Cat}$-operad from the fundamental groupoids functor $\Pi_1: \textbf{Top} \to \textbf{Cat}$ as they are symmetric monoidal functors. And $\Pi_1 (D_2)$ for example is equivalent in $\textbf{Cat}$ to the operad $PaP$ of parenthesized braids. So $PaP$ is an $E_2$-operad in $\textbf{Cat}$, and an $E_2$-algebra in $\textbf{Cat}$ would be a braided monoidal category.
And my question is
i) can we obtain from a $\textbf{Top}$-operad an operad over any symmetric monoidal model category? So that we can always talk about $E_n$-operads and $E_n$-algebras in any symmetric monoidal model category?
ii) Intuitively, $E_n$-algebra in $W$ governs objects in $W$ with $n$-tuply monoidal structures. On each occassion before we talk about $E_n$-algebra in $W$, do we need to know an operad $P$ in $W$ weakly equivalent to $F(D_n)\in W$ (like we have the groupoids operad $PaP$ for the $E_2$-algebra in $\textbf{Cat}$)? If not, how can we talk about $E_n$-algebras in $W$ without knowing an $E_n$-operad $P$ on each time.
iii) In light of understanding $E_n$-algebras in higher categories, how can an operad over higher categories be obtained from a topological operad?