Bernardo is the first to challenge Francisco because Bernardo has seen the ghost, along with Marcellus, and Francisco has not. Bernardo is much more on edge and worried because he fears running into the ghost. Francisco even notes that Bernardo comes "carefully" on his hour: a word which means "on time" but also "cautiously."

We learn, when Bernardo questions him, that Francisco has seen nothing on his watch, not even a mouse. It seems clear that the ghost only shows up on Bernardo's watch, which begins at midnight.

As for the second part of the question, the whole opening setting, on the castle ramparts at midnight, with Bernardo and Marcellus on edge about the "dreaded sight" of the ghost, creates an atmosphere of eeriness and foreboding that sets the tone for the entire play.

(It could be that Bernardo is keyed up, because he knows he might see the ghost again, and that sharpens his reflexes.)