At the left side of the brain is the seat of language and processes in a logical and sequential order. The right side is more visual and processes intuitively, holistically, and randomly.
Both sides of the brain have essentially the same function except for Brocca's areas which controls speech and in Wernicke's area which is essential in the understanding of language.
Yes, the nerves projecting from each hemisphere cross at the medulla, causing the left side of the brain to control movement in the right of the body, and vice versa.