There may be exams during the last week of school (ie, state educational exams that have to happen on a standard date, even if it's inconvenient to the local school district), but I can't think of any school board who would coincide the last day of high school with an exam. Around here, it's usually a half-day anyways. Awards ceremonies usually happen over the last two or three days of school (during the daytime in smaller schools, at nighttime for larger schools), and the students have been dismissed by lunchtime.
If the kid isn't a graduating senior, sometimes they use sports as a disciplinary means during the summertime. They have to show up to football practice, run laps, do push-ups, be there for practice at 6 am before it's 120*, and do all the training stuff of the usual summer training team stuff. We're a rural school district, so I know some years, the football team has helped harvest hay and haul hay bales for local farmers/ranchers as part of their summer physical training.
It's much less certain how much those kids are allowed to play... because they're also the same kids who are so spiteful about their punishment, that they deliberately try to sabotage the team's performance on the field when the time comes for an actual game, in an effort to make the coach look bad.
College--- yeah, that's totally different. You can leave campus anytime you want as soon as you've finished your last exam. But not high school in the US, at least in my part of the world.
