Answer
$a.$ Not necessary.
$b.$ Necessary.
Work Step by Step
$a.$ Not necessarily. We need that $f(-x)=f(x)$ i.e. the graph of the function has to be symmetric with respect to the $y$ axis, but it can intersect it anywhere. The example is $f(x)=x^2+1$. This is an even function but $f(0)=1.$
$b.$ It is necessary. For all $x$ we have $f(-x)=-f(x)$ so for $x=0$ it must be that $f(-0)=-f(0)$ or simply $f(0)=-f(0)$ which is possible only if $f(0)=0.$
