$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)

Fix your favorite compact connected space $X$ - then $X^\omega$ is compact, connected, and homeomorphic to its own square.

The Cantor set is a counter-example to the second and third statement. Note that the Cantor set is homeomorphic to $\{0,1\}^{\mathbb N}$, hence it is homeomorphic to the product with itself.

An infinite set with the smallest topology (exactly two open sets) is a counter-example to the first statement. Martini gives a better counter-example in a comment.

Take the Cantor set. This gives a counterexample for B and C. For A, as Mike said in the comments, take the set $[0,1]^\infty$.