calculus - Confused about differentiability/continuity/partial derivative existence

Ok, so in my notes it says

Prop 1: If a function is differentiable, it will be continuous AND it will also have partial derivatives.

Prop 2: If a function is continuous, or has partial derivatives, or has both, it does not guarantee the function is differentiable.

And the example to follow for prop 2 is:
$f(x,y)=\frac{y^3}{x^2+y^2}$ if $(x,y )\ne (0,0)$

$f(x,y)=0$ if $(x,y)=(0,0)$

$f_x(0,0)=0$

$f_y(0,0)=1$ (how????)

The function is also continuous at $(0,0)$ since $\lim f(x,y)=0$ (using squeeze theorem)

So it says the partial derivatives exist.

My first question is, why is $f_y(0,0)=1$? shouldn't it be $0$? Not that it makes a difference. The partials will exist regardless.

My second question is, it says that this function is not differentiable. How do they know that?

My third question: It says in the calculus textbook, one of the theorems (theorem 8 of chapter 14.4 for stewert's book): If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$. How does this make sense? The example in my notes just said a function can be continuous and have partials, but still not be differentiable