For #2 a polynomial of degree less than 5 is written as $\displaystyle a+bx+cx^2+dx^3$ where $\displaystyle a,b,c,d\in \mathbb{Z}_2$. For each $\displaystyle a,b,c,d$ we have two choices either 0 or 1 since we are working in $\displaystyle \mathbb{Z}_2[x]$. Therefore, there are $\displaystyle 2^4 = 16$ such polynomials. For #3 do you know how to apply the division algorithm?
EDIT: Error! It should have been $\displaystyle a+bx+cx^2+dx^3+ex^5$ and in that case we have $\displaystyle 2^5 =32$.