Do supervisors and other workers have a right to give their employees a hard time?

They are your boss, they can manage how they see fit. If you don't agree with the way they're doing their job, talk to their boss or HR. 

Also depends on what you call a 'hard time'. I worked for bosses that yelled constantly, and that thought what ever I did wasn't good enough. I quit those jobs and found a job with a decent supervisor.  

Only if they want to hold their employees to a high standard.

Supervisors and other workers aren't required to be nice to their subordinates and co-workers. 

You didn't mention the world "bullying" but be mindful that a supervisor or co-workers correcting another employee is not bullying. 

Depends on whether or not you are doing your job...

N.Shadows