Any one of those three is correct- with the understanding that x and are functions of u and v (and vice-versa) they all mean the same thing.

2)I know we can get $\displaystyle \frac{\partial z}{\partial r}=\frac{\partial z}{\partial x} \frac{\partial x}{\partial r}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial r}$and so on for $\displaystyle \frac{\partial z}{\partial \theta}$.

Now lets say we have $\displaystyle z=u(x(r,\theta),y(r,\theta))$

How do I get $\displaystyle \frac{\partial z}{\partial r}$ and $\displaystyle \frac{\partial z}{\partial \theta}$ in terms of $\displaystyle \frac{\partial u}{\partial x}$ and $\displaystyle \frac{\partial u}{\partial y}$?

Do I differentiate both sides of the equation?

If you know x and y as functions, of r and $\displaystyle \theta$, do the differentiations and put them into that equation. For example, if $\displaystyle x= r cos(\theta)$ and $\displaystyle y= r sin(\theta)$, then

$\displaystyle \frac{\partial z}{\partial r}= \frac{\partial z}{\partial x}\frac{\partial x}{\partial r}+ \frac{\partial z}{\partial y}\frac{\partial y}{\partial r}$

$\displaystyle = cos(\theta)\frac{\partial z}{\partial x}+ sin(\theta)\frac{\partial z}{\partial y}$.

Thanks