Can ALU do square roots

Answer 1

No - computing a square root with any degree of accuracy is both a numerical and iterative process. Because of this, it can not be calculated with a combinatorial circuit like an ALU. It requires a sequential circuit, so you at the very least need some memory paired with the ALU to compute a square root.

If you are interested in the numerical methods for calculating square roots, go see the wikipedia page on the topic (check the link).