Here are three questions all related to uniform distribution:

Can one construct a joint density $f(x,y)$ such that the marginal distribution is $U[a,b]$ for $X$, $U[c,d]$ for $Y$ and the correlation between $X$ and $Y$ is $\rho\in(0,1)$? Is there a simple mathematical expression for $f(x,y)$? 

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y]$ is a piecewise linear function of $y$?

The third one is:
Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y, Y\geq \bar{y}]$ (where $\bar{y}$ is some pre-specified number that is within the support of $Y$) is a piecewise linear function of $y$?