Is cocycle condition necesarry for coassociativity of coproduct and why is it called "cocycle condition"? Let $\Delta_0$ and $\Delta$ be coproducts related by
\begin{equation}
\Delta h = \mathcal F \Delta_0 h \mathcal F^{-1}
\end{equation}
where $\mathcal F \in H\otimes H$ is Drinfeld twist and $h \in H$ is just an element of Hopf algebra.
The coproducts should be coassociative, i.e. 
\begin{equation}
(\Delta_0 \otimes 1)\Delta_0h=(1 \otimes \Delta_0)\Delta_0h
\end{equation}
\begin{equation}
(\Delta \otimes 1)\Delta h=(1 \otimes \Delta)\Delta h
\end{equation}
Using $\Delta h = \mathcal F \Delta_0 h \mathcal F^{-1}$ with those two relations yields that
\begin{equation}
\left[
(1 \otimes \mathcal F)(1 \otimes \Delta_0)\mathcal F
\right]^{-1}
\left[
(\mathcal F \otimes 1)(\Delta_0 \otimes 1)\mathcal F
\right]\equiv X
\end{equation}
and its inverse commute with any $(\Delta_0 \otimes 1)\Delta_0 h = (1 \otimes \Delta_0)\Delta_0 h \in H \otimes H \otimes H$.
Does this mean that $X$ has to be proportional to the unity? If so then
\begin{equation}
(\mathcal F \otimes 1)(\Delta_0 \otimes 1)\mathcal F = \lambda
(1 \otimes \mathcal F)(1 \otimes \Delta_0)\mathcal F
\end{equation}
where $\lambda$ is ordinary nonzero number.
Also, will $\lambda$ then necessarily be one? If so, then this is a cocycle condition:
\begin{equation}
(\mathcal F \otimes 1)(\Delta_0 \otimes 1)\mathcal F =
(1 \otimes \mathcal F)(1 \otimes \Delta_0)\mathcal F
\end{equation}
It is obvious that cocycle condition is sufficient condition for coassociativity of $\Delta$ if $\Delta_0$ is coassociative. But is it also necessary condition?
And why is it called "cocycle condition"?
 A: First to say something about why this is a "cocycle condition": The class of quasi-Hopf algebras (coassociativity up to a 3-cocycle) is stable under Drinfeld twists (by 2-cocycle). Chapter 6 of [1] describes how these 2 and 3-cocycles can be enhanced to the so-called non-abelian cohomology.
The paper [2] I found an interesting result related to your first question: For a given 3-cocycle $\Phi$, the authors introduce the notion of a $\Phi$-cocycle as an element $F\in H\otimes H$ for which your $X$ is equal to $\Phi$. For such elements, Proposition 3.2 the Drinfeld twist of $H$ by $F$ is a quasitriangular Hopf algebra.
Because of this Proposition, I think $\Phi$ is not necessarily proportional to $1$ as otherwise the theory of $\Phi$-cocycles would be useless.
It would be nice to have a concrete counterexample. Did you check the original Quantum Group examples by Drinfeld concretely?
[1] Majid, S. Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras. Generalized symmetries in physics (Clausthal, 1993), 13--41, World Sci. Publ., River Edge, NJ, 1994
[2] Gurevich, D.; Majid, Shahn. Braided groups of Hopf algebras obtained by twisting. Pacific J. Math. 162 (1994), no. 1, 27--44
