If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that
Since $W(1/t)\space \sim N(0,1/t)$
that $V(t) \sim t\cdot N(0,1/t)$?
And if so then is $t\cdot N(0,1/t) = N(0,t^2\cdot 1/t) = N(0,t)$?
Yes. In fact $t\mapsto t\cdot W(1/t)$ is itself a Wiener process, strange as it may seem.