Yes. (This is corrected and expanded since the first version.) There are easy examples of simple groups $G$ such that $G\times G$ is isomorphic to a subgroup of $G$. One example is the group of finitely supported permutations of a countable infinite set. Another is the quotient of all permutations of a countably infinite set by the finitely supported ones. Another is the infinite special linear group (direct limit of $SL_n(k)$) of a field.

Yes. (This is corrected and expanded since the first version.) There are easy examples of simple groups $G$ such that $G\times G$ is isomorphic to a subgroup of $G$. One example is the group of finitely supported even permutations of a countable infinite set. Another is the quotient of all permutations of a countably infinite set by the finitely supported ones. Another is the infinite special linear group (direct limit of $SL_n(k)$) of a field.

Yes. Let $G$ be the group of permutations of a countable infinite set, and let $H$ be $G\times G$. $H$ is isomorphic to a subgroup of $G$.

Yes. (This is corrected and expanded since the first version.) There are easy examples of simple groups $G$ such that $G\times G$ is isomorphic to a subgroup of $G$. One example is the group of finitely supported permutations of a countable infinite set. Another is the quotient of all permutations of a countably infinite set by the finitely supported ones. Another is the infinite special linear group (direct limit of $SL_n(k)$) of a field.