Existence of optimal strategy in a choice problem with uncertainty and information structure
Yes, an optimal strategy always exists for any information structure. 

Proof: 1st- the set $\mathcal{Y}$ is finite (thus compact). 2nd- for each state of the world, $v$, $u(y,v)$ is continuous in $y$ using the discrete metric. 3rd- Given the linearity of integrals, for each $t\in\mathcal{T}$ we have that $\int_{\mathcal{V}}u(y,u)P_{T|v}(v)P_V(dv)$ is also continuous in $y$. Therefore Weierstrass theorem ensures the existence of a minimum and a maximum. 

Let $y^*_t$ be one such maximizer when the signal is $t$ (note it can be different for each signal realization, and that the maximizer might not be unique. In the latter case, I arbitrarily choose one for simplicity). Finally, define $P_{Y|T}(y|t)=\mathbb{1}\{y=y^*_t\}$. I.e. to play $y^*_t$ with probability one when the signal is $t$. 

In this setup, existence might only fail if $\mathcal{Y}$ was not compact, or if $u(y,v)$ was not continuous with respect to $y$. Note I said might, even if these two conditions are not present, an optimal might still exist.