What $A_\infty$ operad gives rise to the $A_\infty$ relations? The $A_\infty$ relations 
$$ \sum _{i + j + k = n} (-1)^{\cdots} \mu _ {i + 1 + k} ( \mathbb{1}^{i}, \mu _{j}, \mathbb{1} ^ k) = 0 $$
(for example as defined here) in an $A_\infty$ algebra or an $A_\infty$ category can be explained using trees or Stasheff associahedra.
In the case of the Fukaya category these relations are proven by invoking the innocent fact that every oriented 1-dimensional compact manifold has a trivial (signed) boundary and applying this to the moduli space of gradient flow lines, ref.
Q. Is there an operad which packages this entire structure and not just explains it? That is, is it possible to define an operad whose modules would automatically satisfy the $A_\infty$ relations? Hopefully $A_\infty$ morphisms would then simply be morphism of modules over this operad.
Q. Are the notions of twisted complexes and Karoubi completions also operadic in nature? 
Any references would be very helpful. Thanks.
 A: Yes, there is an operad encoding $A_\infty$-algebras if that is what you're asking.
One simple way of seeing that is that an $A_\infty$-algebra is defined in terms of operations with a certain number of inputs and one output, and that in each relation (e.g. $d(f_1(x)) = f(d(x))$ or $f_1(xy) - f_1(x) f_1(y) = df_2(x,y) \pm f_2(dx,y) \pm f_2(x,dy)$) each variable appears exactly once in each summand. So you can simply take the free operad on the operations, and quotient out by the operadic ideal generated by the relations; almost tautologically, this will give an operad that exactly encodes $A_\infty$-algebras.
But this is not terribly interesting, and I guess what you actually want to know is where this operad comes from? The operad $\mathtt{Ass}$ encoding associative algebras is binary quadratic (it is generated by a binary operation, and the defining relation is quadratic). It is a Koszul operad, so the cobar construction on its Koszul dual $\Omega \mathtt{Ass}^¡$ is a "cofibrant resolution" of $\mathtt{Ass}$. It turns out that $A_\infty = \Omega \mathtt{Ass}^¡$. If you're not familiar with these terms, it means that there is a morphism of operads $\Omega \mathtt{Ass}^¡ \to \mathtt{Ass}$ which:


*

*Induces a Quillen equivalence between the category of associative algebras and the category of $A_\infty$-algebras;

*Roughly speaking, $A_\infty$ satisfies the homotopy transfer theorem: if you've got an $A_\infty$-algebra $A$ and a chain complex $X$ homotopy equivalent to $A$, then you can make $X$ into an $A_\infty$-algebra.


(Infinity-)Morphisms of $A_\infty$-algebras are a bit more complicated. Plain morphisms of algebras over $\Omega \mathtt{Ass}^¡$ are not the morphisms described in the paper you linked; they would be morphisms with only a linear part. You have to look more closely at the operad $A_\infty$ to find what they are.
The operad $A_\infty = \Omega \mathtt{Ass}^¡$ is the cobar construction on the Koszul dual cooperad $\mathtt{Ass}^¡$. It turns out that $\mathtt{Ass}^¡$ is the cooperad encoding shifted associative algebras, $\mathscr{S}^{-1} \mathtt{Ass}^c$. This in turns means that to give $A$ an $A_\infty$-structure, you have to give a square zero codifferential $d_A$ on the cofree shifted coassociative coalgebra $T^c(\Sigma A)$. Then given two such $A_\infty$-algebras $(T^c(\Sigma A), d_A)$ and $(T^c(\Sigma B), d_B)$, an (infinity-)morphism of $A_\infty$-algebras is a morphism of dg-coalgebras $(T^c(\Sigma A), d_A) \to (T^c(\Sigma B), d_B)$.

Hopefully all this helps you understand the operad $A_\infty$ better, and how the story would play out if instead of homotopy associative algebras we were looking at homotopy Lie algebras, homotopy commutative algebras... It wouldn't be too reasonable to make a very detailed answer on math.SE, so I refer you to the excellent book Algebraic Operads by Loday and Vallette, especially Chapter 7 (but you probably need to at least skim the first six chapters to understand it).
