First question: Let $H'$ be a subgroup of $H$ and $K'$ a subgroup of $K$. Is it true that $H'\rtimes K'$ and $H'\times K'$ are subgroup of $H\rtimes K$?
Second question: Let $G=(\mathbb{Z}/p\mathbb{Z})^{n}\rtimes (\mathbb{Z}/p\mathbb{Z})^{m}$. What is the number of subgroups of $G$ isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{2}$?
Third question: Is it true that if $\gcd (|H|,|K|)=1$ then all subgroups of $H\rtimes K$ are of the form $H^{\prime }\rtimes K^{\prime }$.
First question: If it exists, it's a subgroup, but since conjugation by elements of $K^{\prime}$ may take elements of $H^{\prime}$ to elements of $H\setminus H^{\prime}$, it need not exist.
Second question: That depends on the action involved in the semidirect product. For example, consider $2^2\rtimes 2$. If the action is trivial, we get $2^3$ which has seven subgroups isomorphic to $2^2$. If the action is non-trivial, we get $D_4$, which has only two such subgroups.
(Note that $p^n$ is traditional shorthand for $(\mathbb{Z}/p\mathbb{Z})^{n}$.)
Third question: No, since at the very least, if the action is non-trivial, $K$ has conjugates that intersect $H$ trivially and are not contained in $K$.