This, for the most point, contains many valuable points. Very small, incremental changes in the minimum wage don't typically affect the economy-at-large because most workers do not make minimum wage. There is a notion on the left that major cities are full of nothing but minimum wage workers who are dying because they make $7.25 an hour. Which is patently untrue because most people make more than $7.25/hour.
If this is the argument you're making, then it has nothing to do with economics. This is a moral imperative issue and why we're discussing it in the context of economics, I do not know. It would be nice if you would just come out and say that you want this because it's good for a small minority percentage of people that live on the minimum wage.
It's absurd to state, though, that it doesn't affect some small businesses. If Billy Bob in Bucksnort, TN (actual place) owns a hardware store and employs 60 hours/week of minimum wage work to local high school kids, then he spends about $23,000 in very low-skill, entry level work if you go on a minimum wage of $7.25/hr. If President Obama or Mrs. Clinton signs a minimum wage bill into law that increases the minimum wage to $10.50/hr, then Billy Bob incurs an extra $10,000 yearly in payroll expenditures for high school clerks and stockers.
Now, you may look at my above example and scoff and think that $10 grand is a drop in the bucket or that big corporations are able to pay that, no problem. But if you think $10grand/year is a drop in the bucket for a small business, then you are simply wrong. A lot of small businesses would be hit hard by such an increase. The companies that pay people minimum wage would absolutely be affected by this. The Googles and the Apples of the world don't pay people minimum wage anyway, so this won't affect them. But I live in small town TN. I go to little stores and businesses all the time that would have to cut that entry-level job if it meant $10k more in payroll expenditures.
You have to take off the liberal shades for a second and realize that not all businesses are GE or Disney or Google with a backroom full of unused capital. And keep in mind that those businesses are paying most of their people way more than minimum wage anyway so they don't care either way.