[ThE-dArK-lOrD]

For $15)$
Let $P(x,y)$ be the assertion $f(x^3+y^3)=xf(x^2)+y^2f(y)$
$P(x,0),P(0,x)$ give us $f(x^3)=xf(x^2)$ and $f(y^3)=y^2f(y)$ for all $x,y\in \mathbb{R}$
So $P(x,y)$ become $f(x^3+y^3)=f(x^3)+f(y^3)$ for all $x,y\in \mathbb{R}$
This give us $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$
And since $f(x^2)=xf(x)$ for all $x\in \mathbb{R}$
Then $(x+1)(f(x)+f(1))=(x+1)f(x+1)=f(x^2+2x+1)=f(x^2)+2f(x)+f(1)$,
Give us $xf(x)+xf(1)+f(x)=xf(x)+2f(x)$, so $f(x)=xf(1)$ for all $x\in \mathbb{R}$
So $f(x)=cx$ for all $x\in \mathbb{R}$ is only solution

P.S. I think $16$ is really hard, if you want to see solution, I will post later