Clarification on the definition of Bayes Correlated Equilibrium
I think the reason you're having difficulty is that your definition is actually not equivalent to Bergemann and Morris' definition of BCE, except under specific assumptions which include the ones you state regarding independence. However, in general, we do not wish to make these assumptions. (For example, assuming that $t_{-i}$ is independent of $a_i$ conditional on $\theta$ and $t_i$ means that the mediator cannot reveal any additional information about other players' types once they have revealed the state to a player. This seems unnecessarily restrictive.)

In the paper, a BCE induces a probability distribution $\Pr$ that satisfies

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a^\prime_i,a_{-i},\theta) $$

for all $a_i^\prime \in A_i$. If we wish to express this inequality in terms of player $i$'s expected utility given their information, then we divide both sides by $\Pr(a_i,t_i)$. Similarly, from the inequality 

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_{-i},t_{-i},\theta \vert a_i,t_i) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) u_i(a^\prime_i,a_{-i},\theta), $$

we can multiply both sides by $\Pr(a_i,t_i)$ to recover the first expression above. To see how this squares with your calculation, notice that

$$ \begin{align*} \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) \Pr(a_i,t_i) &amp;= \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) \Pr(a_i,t_i) \\
&amp;= \Pr(a_i,a_{-i},t_i,t_{-i},\theta) \\
&amp;= \Pr(a_i,a_{-i}\vert t_i,t_{-i},\theta) \Pr(t_i,t_{-i} \vert \theta) \Pr(\theta)\\
&amp;= \sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta).
\end{align*} $$

In particular, we have that $$ \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) = \frac{\sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta)}{\Pr(a_i,t_i)}. $$ This makes it difficult to get the decomposition you want without making needlessly strong independence assumptions.

When $N=1$, there is only one player, so the equilibrium is (in some sense) degenerate. However, the difference between one-player BCE and one-player BNE is that the BCE set is larger, and includes any BNE. (Indeed, this is true even when $N&gt;1$.)

For your second question:


  a BCE is still intended as mapping from state and signal to a probability distribution over actions. From reading at p.26 of the linked paper, the authors then say "[...] In that case, the set of BCE correspond to joint distribution of actions and states [...]". I'm confused. 


Formally, a BCE is a probability distribution of actions for each realisation of the state and signals. This probability distribution induces a joint distribution over actions and states. Just integrate out $(t_i,t_{-i})$ in $\Pr(a_i,a_{-i},t_i,t_{-i},\theta)$ to get that joint distribution.

Finally, this equilibrium concept is called Bayes Correlated Equilibirum because it extends our definition of correlated equilibrium (defined by Aumann for complete information games) to games of incomplete information, in the same way that Bayes-Nash equilibrium extends Nash equilibrium.