If an employer in Michigan mails paychecks, do the checks have to be in the employee's hands on payday?

No. They have to be postmarked on payday.

If the employee wanted to collect the check in person they could only do so on payday. If the check is to be mailed, the employer mails it on payday and the employee gets it whenever his post office carrier delivers it even if it's a week late.

Yes, the post mark has to be the payday.