Suppose that the continuous random variables X and Y ... I've tried to attempt all these questions myself first but could someone tell me if these are correct? 


The first is okay.   You have found $f_Y(y)$, and $f_X(x)$, although you should also make note of their supports.
The second is not okay.   You are after: $f_{Y\mid X}(y\mid x) ~=~ \dfrac{f_{X,Y}(x,y)}{f_X(x)}$
The third is really not okay.   Hints: $$\begin{align}\mathsf P(Y>t) ~=~& {\displaystyle\int_{t}^1} f_{Y}(y)\operatorname d y \\[2ex]~\mathsf P(Y>t\mid X=s) ~=~& {\displaystyle\int_{t}^1} f_{Y\mid X}(y\mid s)\operatorname d y\end{align}$$
