It drives me crazy how many employers pay minimum knowing people can't live off of that
Yes, the exploitation of labor is the act of treating workers unfairly for ones own benefit.  There are many companies that do this, but not all do.
Companies that make huge profits should give back to the ones making them that money
I agree.  The social relationship between the workers and the employers should be based on hard work by the workers, and fairness by the employers.  It boosts morale of all workers to see that both employees and employers are on the same team for each one's benefit. 
I have worked all types of jobs and I completely agree. It can really make or break an employee
The minimum wage, to me, should not even be necessary - not because employers should pay their employees less, but rather that employers should value their workers enough to pay reasonable amounts as they can.  I am bothered by companies that make billions in profits, but can only "afford" to pay minimum wage.  If that is true, then there is something wrong with the tax system.
Yes, I think they call it social or moral responsibility. There's obviously a lack of respect going on somewhere
It is a poor view of fellow man for sure.  Granted, there are difference of opinion about the minimum wage, and even different jurisdictions for it; but ultimately, I feel that the moral responsibility falls on the employer, not the government.