1) Why should there be a national minimum wage?
Sometimes, the answer is that the best possible law is too complicated to actually work. My opinion is that the need for the principle of a minimum wage is very clear, and it is up to legislators to work it out. (God help us all.)
2) I would argue with your first point above. If a company makes a profit, then the work performed by all the employees is more valuable than the wages paid to the all the employees.
Exactly. The example is simplified. I know some people who make a whole lot more than minimum wage whose true value to the company is somewhat less than zero.
And of course, the services of one, single, employee are rarely worth anything. It's the total work performed by the whole team of employees.
What should an employee be paid if his services are worth less than the minimum wage?
He should be fired. If his lack of value is due to his problem, for example he is too lazy to work, get rid of him and find someone not so lazy. If his labor is inherently of so little value that you cannot afford to pay him a decent salary, as defined previously, then you should not be allowed to profit from his labor.
Now, you would not demand that a business pay a minimum price for its phone service;
No, but I would demand the phone company pay its employees minimum wage, which they would then pass on to the business that uses its service. Indirectly, then, I am demanding that the business pay a minimum price for its phone service.
So why should it pay $5.15 to an employee who's only worth $1.25?
I believe that every person not suffering from a handicap is capable of doing labor that is enough to keep him alive at a decent level. In other words, I believe that every non-handicapped person is capable of producing something that is worth a fair minimum wage. If you, as an employer, cannot figure out a way to harness that person's talent sufficiently for him to make enough to feed himself, then you aren't a very good businessman, and you should go get a job, and release that person into the employment pool where someone else can harness them more effectively.
For handicapped people, we provide a subsidy to employers so that we, as a society, shoulder the burden instead of sticking it to the employer who is trying to do what he can for a handicapped person.