Robust predictions in single-agent decision problem with uncertainty
1) Given that $\mathcal{Y}$ is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about $\mathcal{V}$ that could be formed (as a posterior belief) after observing some signal and having a prior $P_V$. 

To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be $P_Y^*$ (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define $P_{Y,V}=P_Y^*P_V$, then such a distribution constitutes a BCE.

Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in $\mathcal{Y}$ that is the best action regardless of any extra information you might receive. 

2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, $P_V$. If you want to  put it in terms of the two expressions you have, then $P_{Y,V}$ must  be a product distribution, i.e $P_{Y,V}(y,v)=P_Y(y)\cdot P_V(v)$ for some $P_Y(y)\in\Delta(\mathcal{Y})$ (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down $P_Y$. In fact, using the notation from the previous bullet point, it should be equal to $P_Y^*$.