I agree with your criticism of the unemployment calculation. However, the issue that you are missing with regard to the inverse relationship between unemployment and minimum wage is that not every potential employees time is worth the same amount to an employer.
If the minimum is $8/hr, but my labor is only worth $6/hr, then I will be unemployed. My labor might not be worth that extra $2/hr because of my poor education, work ethic, language skills, personality, etc. However that doesn't change the fact that a company would lose money by hiring me.
Also, because of other costs like insurance, retirement benefits, and human resources costs that many governments require you to pay on behalf full-time employees, but not part time employees, you can frequently hire 3 or more $4/hr employees in place of a single $8/hr employee and for less in total cost. Fringe, the HR term for employee costs in addition to salary, can be anywhere from 30-60% of base salary for a full time employee.
I am speaking from 1st hand experience. The research lab I worked in hired 2 temporary workers for less than the cost of a single permanent employee for basic tasks. I wanted to make one of the temporary employees permanent, but we just could not afford it