I agree with the sentiment that there are too many variables for long term studies to cut through.
As for the rest, well, I don't know. I think the data is ambiguous and inconclusive for at least the short term. And possibly even the long term, because you get into factors like "learned unemployment," where if someone falls out of the labor pool for too long, they are less likely to reenter it even if the economy picks up again, whether due to employer discrimination (this is well-documented) or because of other factors (e.g., the unemployed person becomes clinically depressed, or makes a false claim of disability to get by once welfare checks run out). On the other hand, if wages are so low that minimum wage is not enough to support someone, then that someone may decide to keep remaining unemployed rather than work for barely any more money than they get from welfare/unemployment. They could then add to the pool of people who have been out of work for so long that they have a hard time getting back to work. This really happens, by the way, especially if someone needs a car and fuel to get to work, as the car payments and fuel can eat up a big chunk of someone's minimum wage. So they continue to not-work.
Someone else beat me to it, but poor people tend to spend more of their earnings than rich people, for obvious reasons. So that minimum wage gets recirculated into the local economy, which may improve the sales revenue and profits of the employer. That doesn't always happen, though, especially with commodity products that people can easily order online. But there remains some of this effect which can offset any negative impact from minimum wage.
For this and other reasons (some of which were laid out in the blog), small increases in wages it would appear that the minimum wage does not always produce a statistically significant negative effect on employment in the short term. The various studies' collective results are inconclusive.