Yes. Let $a_1$ be a generator of $K_3$ over $K_1$ and let $a_2$, $a_3$, and $a_4$ be the other three roots of the polynomial of $a_1$. Then say that $a_2$ is the other root of the polynomial of $a_1$ over $K_2$. Then The Galois group of $K_3/K_1$ acts on the partitioned set $\{\{a_1,a_2\},\{a_3,a_4\}\}$, and is therefore a transitive subgroup of $D_4$, either $D_4$ itself or $C_4$ or $C_2 \times C_2$.

I think that the following is a more general fact that can be proven in the same way. If $K_2/K_1$ is separable and its Galois closure has Galois group $G$, and if $K_3/K_2$ is separable and its Galois closure has Galois group $H$, then the Galois group of the Galois closure of $K_3/K_1$ is a subgroup of a wreath product of $G$ and $H$. If you take a longer chain of extensions you get an iterated wreath product.

Yes. Let $a_1$ be a generator of $K_3$ over $K_1$ and let $a_2$, $a_3$, and $a_4$ be the other three roots of the polynomial of $a_1$. Then say that $a_2$ is the other root of the polynomial of $a_1$ over $K_2$. Then The Galois group of $K_4/K_1$ acts on the partitioned set $\{\{a_1,a_2\},\{a_3,a_4\}\}$, and is therefore a transitive subgroup of $D_4$, either $D_4$ itself or $C_4$ or $C_2 \times C_2$.

I think that the following is a more general fact that can be proven in the same way. If $K_2/K_1$ is separable and its Galois closure has Galois group $G$, and if $K_3/K_2$ is separable and its Galois closure has Galois group $H$, then the Galois group of the Galois closure of $K_3/K_1$ is a subgroup of a wreath product of $G$ and $H$. If you take a longer chain of extensions you get an iterated wreath product.