Is there a place to see staff members (HC, OC, DC) that are on the market or to make offers for them? Is this available only at certain times of the season? Can you fire a coach in mid-season?

This is done during staff hiring phases. For HC on day 1 you can offer to public candidates (stealing them from another school) or private candidates (only available to you). For OC/DC or for HC day 2 it's just private.