Yes, in general $L$ need not be a prime knot, if $K$ is a prime knot, given a knot epimorphism $f\colon \pi K \to \pi L$, see section $4$ of D.S. Silver and W. Whitten, Knot group epimorphisms II (which is referring to part I for an example).
There it is also discussed which properties of $K$ are inherited by $L$, given a knot group epimorphisms $f\colon \pi K \to \pi L$. For example, if $K$ is alternating, $L$ need not be alternating in general.

Yes, in general $L$ need not be a prime knot, if $K$ is a prime knot, given a knot epimorphism $f\colon \pi K \to \pi L$, see section $4$ of D.S. Silver and W. Whitten, Knot group epimorphisms II (which is referring to part I for an example).
There it is also discussed which properties of $K$ are inherited by $L$, given a knot group epimorphisms $f\colon \pi K \to \pi L$. For example, if $K$ is alternating, $L$ need not be alternating in general. On the other hand, if, say, $K$ is a torus knot, then $L$ is, too.

No, in general $L$ need not be a prime knot, if $K$ is a prime knot, given a knot epimorphism $f\colon \pi K \to \pi L$, see section $4$ of D.S. Silver and W. Whitten, Knot group epimorphisms II (which is referring to part I for an example).
There it is also discussed which properties of $K$ are inherited by $L$, given a knot group epimorphisms $f\colon \pi K \to \pi L$. For example, if $K$ is alternating, $L$ need not be alternating in general.

Yes, in general $L$ need not be a prime knot, if $K$ is a prime knot, given a knot epimorphism $f\colon \pi K \to \pi L$, see section $4$ of D.S. Silver and W. Whitten, Knot group epimorphisms II (which is referring to part I for an example).
There it is also discussed which properties of $K$ are inherited by $L$, given a knot group epimorphisms $f\colon \pi K \to \pi L$. For example, if $K$ is alternating, $L$ need not be alternating in general.